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Dilations and similarity

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5138948
Two circular wall mirrors have different sizes. The smaller mirror has radius \(r_1=12\,\text{in.}\), and the larger mirror has twice the radius, \(r_2=24\,\text{in.}\). a) Find the circumference of each mirror. b) Find the area of each mirror. c) By what factor does area change when radius doubles? Briefly explain.

Hints

- Use the circumference and area formulas for a circle. - Which formula contains the radius squared? - Compare the two radii and the two areas as ratios.

Solution

1. The smaller circumference is \(C_1=2\pi(12\,\text{in.})=24\pi\,\text{in.}\approx75.40\,\text{in.}\). 2. The larger circumference is \(C_2=2\pi(24\,\text{in.})=48\pi\,\text{in.}\approx150.80\,\text{in.}\). 3. The smaller area is \(A_1=\pi(12\,\text{in.})^2=144\pi\,\text{in.}^2\approx452.39\,\text{in.}^2\). 4. The larger area is \(A_2=\pi(24\,\text{in.})^2=576\pi\,\text{in.}^2\approx1809.56\,\text{in.}^2\). 5. The area factor is \(\frac{A_2}{A_1}=\frac{576\pi}{144\pi}=4\), because \((2r)^2=4r^2\).

Answer

a) \(C_1\approx75.40\,\text{in.}\); \(C_2\approx150.80\,\text{in.}\) b) \(A_1\approx452.39\,\text{in.}^2\); \(A_2\approx1809.56\,\text{in.}^2\) c) The area is multiplied by \(4\).
5139068
A circle has area \(A\) and circumference \(C\). A new circle is created with \(4\) times the original area. Determine how the new circumference compares with \(C\). Justify your answer by comparing the formulas.

Hints

- How must the radius change for area to be multiplied by \(4\)? - Is the area-radius relationship linear or quadratic? - How does circumference depend on radius?

Solution

1. If the new area is \(4A\), then \(\pi r_{\text{new}}^2=4\pi r^2\). 2. Dividing by \(\pi\) and taking the positive square root gives \(r_{\text{new}}=2r\). 3. The new circumference is \(C_{\text{new}}=2\pi(2r)=2(2\pi r)=2C\).

Answer

The circumference doubles: \(C_{\text{new}}=2C\).
5140418
A rectangular billboard is enlarged proportionally. Its area increases from \(2\,\text{m}^2\) to \(12.5\,\text{m}^2\). Find the factor by which each side length was enlarged. Then state the factor by which the perimeter changed.

Hints

- How is an area scale factor related to a linear scale factor? - Is perimeter a linear measurement or an area measurement?

Solution

1. The area factor is \(\frac{12.5}{2}=6.25\). 2. If the linear scale factor is \(k\), then the area factor is \(k^2\). Thus, \(k=\sqrt{6.25}=2.5\). 3. Perimeter is a linear measurement, so it changes by the same factor, \(2.5\).

Answer

Each side length was multiplied by \(2.5\), and the perimeter was also multiplied by \(2.5\).
5140438
A square photo is reduced for a brochure. The area of the reduced photo will be \(64\%\) of the original area. a) By what factor \(k\) should each side length be multiplied? b) By what percent does each side length decrease?

Hints

- Write \(64\%\) as a decimal. - The area factor is the square of the side-length factor. - Compare the new side-length percent with \(100\%\).

Solution

1. The area factor is \(0.64\), so \(k^2=0.64\). 2. Thus, \(k=\sqrt{0.64}=0.8\). 3. A factor of \(0.8\) means the new side length is \(80\%\) of the original, so the decrease is \(100\%-80\%=20\%\).

Answer

a) \(k=0.8\) b) The side length decreases by \(20\%\).
5147918
Two rectangles are similar when the ratios of their corresponding side lengths are equal. a) A photograph measures \(10\,\text{cm} \times 15\,\text{cm}\). A poster of the same image measures \(40\,\text{cm} \times 60\,\text{cm}\). Use calculations to determine whether the two rectangles are similar. b) International A-series paper sizes are designed so that the ratio of the longer side to the shorter side is always \(\sqrt{2} \approx 1.414\). An A4 sheet measures \(21.0\,\text{cm} \times 29.7\,\text{cm}\), and an A5 sheet measures \(14.8\,\text{cm} \times 21.0\,\text{cm}\). Find the ratio of length to width for each sheet. Based on the rounded measurements given, decide whether the rectangles are exactly similar or only approximately similar.

Hints

- What must be true about corresponding side-length ratios when one figure is a scaled copy of another? - How do you calculate the ratio of two numbers? - Do rounded measurements always produce exactly equal ratios?

Solution

1. For part a, the photograph has side-length ratio \(\frac{15}{10}=1.5\), and the poster has side-length ratio \(\frac{60}{40}=1.5\). Because the ratios are equal, the rectangles are similar. 2. For part b, the A4 ratio is \(\frac{29.7}{21.0}\approx 1.4143\). The A5 ratio is \(\frac{21.0}{14.8}\approx 1.4189\). 3. The ratios from the rounded measurements are not equal, so the two measured rectangles are not exactly similar. They are approximately similar. Ideal A-series sheets are exactly similar because each has side-length ratio \(\sqrt{2}:1\).

Answer

a) Yes. Both rectangles have a side-length ratio of \(1.5\). b) The A4 ratio is approximately \(1.414\), and the A5 ratio is approximately \(1.419\). Using the rounded measurements, the sheets are only approximately similar. Ideal A-series sheets are exactly similar.
5147938
Two geometric figures are similar when corresponding angles have equal measures and corresponding side lengths are proportional. a) Explain why all squares are similar. b) Determine whether all rhombuses are similar. Give a counterexample or justify your conclusion.

Hints

- What angle and side properties do all squares share? - What feature of a rhombus can change even though all four sides remain congruent?

Solution

1. Every square has four \(90^\circ\) angles. If two squares have side lengths \(a\) and \(b\), every pair of corresponding sides has the same ratio \(\frac{b}{a}\). Therefore, all squares are similar. 2. A rhombus has four congruent sides, but its angle measures can vary. For example, a rhombus with angles \(60^\circ\) and \(120^\circ\) is not similar to a rhombus with angles \(80^\circ\) and \(100^\circ\). Therefore, not all rhombuses are similar.

Answer

a) All squares are similar because corresponding angles are congruent and corresponding sides are proportional. b) No. Rhombuses can have different angle measures, so they are not always similar.
5148008
The radius of a circle is increased by \(25\%\). By what percent does the area of the circle increase?

Hints

- Express the new radius as a multiple of the original radius. - The area of a circle depends on \(r^2\). - Subtract \(100\%\) from the new area percent.

Solution

1. Increasing the radius by \(25\%\) gives a linear scale factor of \(k=1.25\). 2. Area scales by the square of the linear factor: \(k^2=1.25^2=1.5625\). 3. The new area is \(156.25\%\) of the original area, so the increase is \(56.25\%\).

Answer

The area increases by \(56.25\%\).
5148028
An architect creates two scale drawings of the same park. The first drawing uses a scale of \(1:500\), and the second, more detailed drawing uses a scale of \(1:200\). By what factor is the area of a lawn on the second drawing greater than its area on the first drawing?

Hints

- Compare how a real-world length is represented at the two scales. - Once you find the linear factor, square it to find the area factor.

Solution

1. The linear scale factor from the first drawing to the second is \(k=\frac{500}{200}=2.5\). 2. Areas scale by the square of the linear factor, so the area factor is \(k^2=2.5^2=6.25\).

Answer

The lawn’s area on the second drawing is \(6.25\) times its area on the first drawing.
5148038
For each pair of geometric figures, decide whether the figures are always similar. Briefly justify your answer. a) Two regular hexagons b) Two isosceles trapezoids

Hints

- What does similarity require about corresponding angles? - Does the name of each figure completely determine its shape, or can its angle measures vary? - Do all regular polygons with the same number of sides have the same angle measures?

Solution

1. Similar figures have congruent corresponding angles and proportional corresponding side lengths. 2. Every regular hexagon has six congruent sides and six \(120^\circ\) angles. Between any two regular hexagons, all corresponding side lengths have the same scale factor. Therefore, two regular hexagons are always similar. 3. Isosceles trapezoids have congruent base-angle pairs, but the angle measures can vary from one trapezoid to another. The ratio of the two base lengths can also vary. Therefore, two isosceles trapezoids are not always similar.

Answer

a) Yes. Two regular hexagons are always similar because their corresponding angles are congruent and their corresponding sides are proportional. b) No. Isosceles trapezoids can have different angle measures and different side-length ratios.
5153918
A technical drawing is enlarged using a constant scale factor \(k\). Complete the table. <table> <tr><th>Original length \(l\) in \(\text{cm}\)</th><th>Enlarged length \(L\) in \(\text{cm}\)</th><th>Scale factor \(k=\frac{L}{l}\)</th></tr> <tr><td>\(5.5\)</td><td>\(13.75\)</td><td>?</td></tr> <tr><td>?</td><td>\(21.25\)</td><td>?</td></tr> <tr><td>\(12\)</td><td>?</td><td>?</td></tr> </table>

Hints

- Use the two known values in the first row to find the constant ratio. - The scale factor is the same in every row. - Rearrange \(k=\frac{L}{l}\) to find a missing length.

Solution

1. Use the first row to find the constant scale factor: \(k=\frac{13.75}{5.5}=2.5\). 2. In the second row, \(l=21.25\div2.5=8.5\,\text{cm}\). 3. In the third row, \(L=12\cdot2.5=30\,\text{cm}\). 4. The value of \(k\) is \(2.5\) in every row.

Answer

The completed table is: <table> <tr><th>Original length \(l\) in \(\text{cm}\)</th><th>Enlarged length \(L\) in \(\text{cm}\)</th><th>Scale factor \(k=\frac{L}{l}\)</th></tr> <tr><td>\(5.5\)</td><td>\(13.75\)</td><td>\(2.5\)</td></tr> <tr><td>\(8.5\)</td><td>\(21.25\)</td><td>\(2.5\)</td></tr> <tr><td>\(12\)</td><td>\(30\)</td><td>\(2.5\)</td></tr> </table>
5153978
A logo has an area of \(80\,\text{cm}^2\). It is enlarged proportionally using a scale factor of \(k=1.5\). Find the area of the enlarged logo.

Hints

- How does area change when both dimensions use the same scale factor? - Square the scale factor before multiplying by the original area.

Solution

1. Area scales by the square of the linear factor: \(A_{\text{new}}=80\cdot1.5^2\). 2. Since \(1.5^2=2.25\), \(A_{\text{new}}=80\cdot2.25=180\,\text{cm}^2\).

Answer

\(180\,\text{cm}^2\)
5331398
Triangle \(ABC\) is shown on the coordinate plane. Dilate the triangle from the origin by a scale factor of \(0.5\). a) Find the coordinates of the image points \(A'\), \(B'\), and \(C'\). b) How do the side lengths of the image compare with the side lengths of the original triangle?
Figure for problem 533139

Hints

- First read the coordinates of each vertex from the graph. - A dilation from the origin multiplies both coordinates of each point by the scale factor. - A dilation multiplies every length by the scale factor.

Solution

1. Read the original coordinates from the graph: \(A(2, 2)\), \(B(8, 2)\), and \(C(4, 6)\). 2. Multiply each coordinate by \(0.5\): \(A'(2 \cdot 0.5, 2 \cdot 0.5)=A'(1, 1)\) \(B'(8 \cdot 0.5, 2 \cdot 0.5)=B'(4, 1)\) \(C'(4 \cdot 0.5, 6 \cdot 0.5)=C'(2, 3)\) 3. A dilation with scale factor \(0.5\) multiplies every side length by \(0.5\), so each image side is half the length of its corresponding original side.

Answer

a) \(A'(1, 1)\), \(B'(4, 1)\), and \(C'(2, 3)\) b) Each side length of the image is \(0.5\) times the corresponding original side length.
5364278
Rectangles \(R_1\) and \(R_2\) are similar. Rectangle \(R_1\) has side lengths \(a_1=6\,\text{cm}\) and \(b_1=4\,\text{cm}\). The longer side of the larger rectangle \(R_2\) is \(a_2=15\,\text{cm}\). a) Find the length of the shorter side \(b_2\) of rectangle \(R_2\). b) Explain what it means for two geometric figures to be similar. Include corresponding side-length ratios and corresponding angles in your explanation.
Figure for problem 536427

Hints

- What remains unchanged about a figure’s shape under a dilation? - What must happen to every side length to preserve the shape? - How can you find the scale factor from the two known corresponding sides?

Solution

1. Because the rectangles are similar, corresponding side lengths have equal ratios: \(\frac{a_2}{a_1}=\frac{b_2}{b_1}\). Substituting gives \(\frac{15\,\text{cm}}{6\,\text{cm}}=\frac{b_2}{4\,\text{cm}}\). The scale factor is \(k=2.5\), so \(b_2=4\,\text{cm}\cdot2.5=10\,\text{cm}\). 2. Two figures are similar when one can be obtained from the other by a dilation, possibly followed by rigid motions. Corresponding angles are congruent, and all corresponding side lengths have the same ratio, equal to the scale factor.

Answer

a) \(b_2=10\,\text{cm}\) b) Similar figures have congruent corresponding angles and proportional corresponding side lengths.
5364328
The grid shows three right triangles labeled I, II, and III. Each grid square has a side length of \(1\) unit. Determine which triangles are similar. Justify your answer by comparing the ratios of their leg lengths.
Figure for problem 536432

Hints

- What conditions make two geometric figures similar? - What angle do all three triangles have in common? - Compare the ratios of corresponding leg lengths. - Count grid squares to find the horizontal and vertical leg lengths.

Solution

1. Count the horizontal and vertical leg lengths: - Triangle I has legs of \(3\) units and \(2\) units. - Triangle II has legs of \(6\) units and \(4\) units. - Triangle III has legs of \(3\) units and \(3\) units. 2. Compare corresponding leg lengths. From triangle I to triangle II, \(\frac{6}{3}=2\) and \(\frac{4}{2}=2\), so the triangles are similar. 3. From triangle I to triangle III, one ratio is \(\frac{3}{3}=1\), while the other is \(\frac{3}{2}=1.5\), so they are not similar. 4. From triangle II to triangle III, one possible ratio is \(\frac{6}{3}=2\), while the other is \(\frac{4}{3}\ne2\), so they are not similar.

Answer

Triangles I and II are similar because both corresponding leg lengths have scale factor \(2\). Triangle III is not similar to either of the other triangles.
5126868
Two circular pizzas are compared. Pizza A has circumference \(40\,\text{in.}\), and Pizza B has circumference \(50\,\text{in.}\). By what percent is Pizza B's area greater than Pizza A's area? Explain why you can find the answer without calculating either radius explicitly.

Hints

- What is the scale factor from one circumference to the other? - How do corresponding lengths scale in similar figures? - How do areas scale compared with lengths?

Solution

1. The circumference scale factor is \(\frac{50}{40}=1.25\). 2. Because circumference is proportional to radius, the radius scale factor is also \(1.25\). 3. Area changes by the square of the radius scale factor, so \(\frac{A_B}{A_A}=1.25^2=1.5625\). 4. The percent increase is \((1.5625-1)\cdot100\%=56.25\%\). 5. No radii are needed because all circles are similar, and their areas are proportional to the squares of corresponding lengths, including their circumferences.

Answer

Pizza B's area is \(56.25\%\) greater than Pizza A's area.
5138478
A landscaper has two circular flower beds. Bed A has diameter \(4\,\text{ft}\). Bed B has twice the circumference of Bed A. a) What is the diameter of Bed B? b) Find the area of each bed, rounded to the nearest hundredth. What percent of Bed B's area is Bed A's area?

Hints

- What happens to diameter when circumference doubles? - How does doubling a radius affect area? - Divide the smaller area by the larger area to find the requested percent.

Solution

1. Bed A has circumference \(C_A=\pi(4\,\text{ft})=4\pi\,\text{ft}\). 2. Bed B has circumference \(C_B=2C_A=8\pi\,\text{ft}\), so its diameter is \(d_B=\frac{8\pi}{\pi}\,\text{ft}=8\,\text{ft}\). 3. Bed A has radius \(2\,\text{ft}\), so \(A_A=\pi(2\,\text{ft})^2=4\pi\,\text{ft}^2\approx12.57\,\text{ft}^2\). 4. Bed B has radius \(4\,\text{ft}\), so \(A_B=\pi(4\,\text{ft})^2=16\pi\,\text{ft}^2\approx50.27\,\text{ft}^2\). 5. The percent is \(\frac{A_A}{A_B}\cdot100\%=\frac{4\pi}{16\pi}\cdot100\%=25\%\).

Answer

a) \(8\,\text{ft}\) b) \(A_A\approx12.57\,\text{ft}^2\), \(A_B\approx50.27\,\text{ft}^2\), and Bed A is \(25\%\) of Bed B's area.
5138628
A circular flower bed will be enlarged. A landscaper considers two plans: Plan A: Double the radius. Plan B: Double the circumference. Determine how the area changes under each plan. What do you notice? Justify your answer generally.

Hints

- How does area change when radius is multiplied by a factor? - If circumference doubles, what happens to radius? - Substitute the changed measurements into the formulas and simplify.

Solution

1. Under Plan A, the new radius is \(2r\), so the new area is \(\pi(2r)^2=4\pi r^2\). The area is multiplied by \(4\). 2. Circumference is proportional to radius because \(C=2\pi r\). Therefore, doubling the circumference under Plan B also doubles the radius. 3. Plan B gives the same new area, \(4\pi r^2\). 4. Both plans multiply the area by \(4\).

Answer

Under both plans, the area is multiplied by \(4\).
5138668
Circle \(K_2\) has a circumference that is \(20\%\) greater than the circumference of circle \(K_1\). By what percent is the area of \(K_2\) greater than the area of \(K_1\)?

Hints

- What multiplication factor represents a \(20\%\) increase? - How does the circumference factor affect the radius? - Square the radius factor to obtain the area factor. - Convert the final factor to a percent increase.

Solution

1. A \(20\%\) circumference increase gives \(C_2=1.2C_1\). 2. Since circumference is proportional to radius, \(r_2=1.2r_1\). 3. Therefore, \(A_2=\pi(1.2r_1)^2=1.44\pi r_1^2=1.44A_1\). 4. An area factor of \(1.44\) represents a \(44\%\) increase.

Answer

\(44\%\) greater
5139058
A circular advertising banner has circumference \(9.42\,\text{ft}\). A second banner will have \(4\) times the area of the first. a) Find the radius of the first banner. b) What radius must the second banner have? Explain the relationship between the radius change and the area change.

Hints

- Use circumference to find the first radius. - What radius scale factor produces an area scale factor of \(4\)? - Area depends on the square of the radius.

Solution

1. The first radius is \(r_1=\frac{9.42\,\text{ft}}{2\pi}\approx1.50\,\text{ft}\). 2. An area factor of \(4\) requires a radius factor of \(\sqrt{4}=2\). 3. Therefore, \(r_2=2r_1\approx3.00\,\text{ft}\). 4. This works because \(A=\pi r^2\), so doubling radius multiplies area by \(2^2=4\).

Answer

a) Approximately \(1.50\,\text{ft}\) b) Approximately \(3.00\,\text{ft}\); multiplying the radius by \(2\) multiplies the area by \(4\).
5139078
Two circles, \(K_1\) and \(K_2\), have diameters \(d_1\) and \(d_2\). The diameter of \(K_2\) is three times the diameter of \(K_1\). a) What is the ratio of the circumferences \(C_1:C_2\)? b) What percent of the area of \(K_2\) is the area of \(K_1\)?

Hints

- How does tripling a number affect its square? - Which formula directly relates diameter and circumference? - If one area is \(9\) times another area, what fraction of the larger area is the smaller area?

Solution

1. Since \(d_2=3d_1\) and \(C=\pi d\), \(C_2=\pi(3d_1)=3C_1\). Therefore, \(C_1:C_2=1:3\). 2. Since \(A=\frac{\pi}{4}d^2\), \(A_2=\frac{\pi}{4}(3d_1)^2=9A_1\). 3. Thus, \(\frac{A_1}{A_2}=\frac{1}{9}\approx0.1111\), or about \(11.11\%\).

Answer

a) \(C_1:C_2=1:3\) b) The area of \(K_1\) is about \(11.11\%\) of the area of \(K_2\).
5141088
Investigate how a circle changes when its radius is tripled. a) Find the circumference and area of a circle with \(r_1=2\,\text{cm}\) and of a circle with \(r_2=6\,\text{cm}\). Round to the nearest tenth. b) Compare your results. By what factor did the circumference increase? By what factor did the area increase? Briefly explain using the formulas.

Hints

- Substitute the tripled radius into the circumference formula. - Substitute the tripled radius into the area formula and pay attention to the exponent. - Divide each new value by its corresponding original value.

Solution

1. For \(r_1=2\,\text{cm}\), \(C_1=2\pi\cdot2\,\text{cm}\approx12.6\,\text{cm}\) and \(A_1=\pi\cdot2^2\,\text{cm}^2\approx12.6\,\text{cm}^2\). 2. For \(r_2=6\,\text{cm}\), \(C_2=2\pi\cdot6\,\text{cm}\approx37.7\,\text{cm}\) and \(A_2=\pi\cdot6^2\,\text{cm}^2\approx113.1\,\text{cm}^2\). 3. Because \(C=2\pi r\), tripling \(r\) triples \(C\). 4. Because \(A=\pi r^2\), tripling \(r\) multiplies \(A\) by \(3^2=9\).

Answer

a) For \(r_1=2\,\text{cm}\), \(C_1\approx12.6\,\text{cm}\) and \(A_1\approx12.6\,\text{cm}^2\). For \(r_2=6\,\text{cm}\), \(C_2\approx37.7\,\text{cm}\) and \(A_2\approx113.1\,\text{cm}^2\). b) The circumference is multiplied by \(3\), and the area is multiplied by \(9\).
5141178
A designer is resizing a circular logo. a) First, the designer triples the radius of the original logo. By what factor does the area increase? b) Next, the designer wants a version whose area is exactly \(9\) times the original area. How should its circumference compare with the original circumference?

Hints

- What role does the exponent play in the area formula? - How can you use an area scale factor to find the radius scale factor? - How is circumference related to radius?

Solution

1. Area is proportional to the square of the radius: \(A=\pi r^2\). Tripling the radius multiplies the area by \(3^2=9\). 2. If the new area is \(9A\), then the radius scale factor is \(\sqrt{9}=3\). 3. Since circumference is proportional to radius, a radius scale factor of \(3\) gives a circumference scale factor of \(3\).

Answer

a) The area increases by a factor of \(9\). b) The circumference must be tripled.
5147968
A rectangular photo is \(10\,\text{cm}\) wide and \(15\,\text{cm}\) long. a) A similar poster will be \(60\,\text{cm}\) wide. Find its length. b) A different similar poster will have nine times the area of the original photo. Find the scale factor \(k\) and the new dimensions.

Hints

- Similar rectangles use the same scale factor for both dimensions. - How is an area factor related to a linear scale factor? - Apply the linear factor to both original dimensions.

Solution

1. For part a), the width scale factor is \(\frac{60}{10}=6\). The length is \(15\cdot6=90\,\text{cm}\). 2. For part b), the area factor is \(9\), so \(k^2=9\) and \(k=3\). 3. The new dimensions are \(10\cdot3=30\,\text{cm}\) and \(15\cdot3=45\,\text{cm}\).

Answer

a) \(90\,\text{cm}\) b) \(k=3\); the dimensions are \(30\,\text{cm}\) by \(45\,\text{cm}\).
5148058
Two parallelograms each have adjacent side lengths in the ratio \(2\) to \(3\). Is this information enough to determine that the parallelograms are similar? Explain your reasoning and state the conditions required for two polygons to be similar.

Hints

- Can two parallelograms have the same side lengths but different amounts of slant? - What two conditions must be satisfied for polygons to be similar? - Is checking only side lengths enough for quadrilaterals? Compare a square and a non-square rhombus.

Solution

1. Two polygons are similar when all corresponding angles are congruent and all corresponding side lengths are proportional. 2. The given ratio shows that the side lengths within each parallelogram have the same proportion. For example, one parallelogram could have adjacent sides of \(2\,\text{cm}\) and \(3\,\text{cm}\), while another has adjacent sides of \(4\,\text{cm}\) and \(6\,\text{cm}\). 3. Nothing is given about the angle measures. One parallelogram could be a rectangle with \(90^\circ\) angles, while the other could have angles of \(45^\circ\) and \(135^\circ\). 4. Because corresponding angles are not guaranteed to be congruent, the side-length ratio alone is not enough to establish similarity.

Answer

No. Similar polygons must have congruent corresponding angles and proportional corresponding side lengths. Parallelograms can have the same side-length ratio but different angle measures, so similarity is not guaranteed.
5148078
Two rhombuses each have side length \(s=5\,\text{cm}\). a) Explain why this information alone is not enough to conclude that the rhombuses are similar. b) What additional condition would guarantee that the rhombuses are similar?

Hints

- Imagine a rhombus made from four hinged rods. What can change while the rod lengths stay fixed? - Are proportional side lengths alone enough to prove that polygons are similar? - What role do corresponding angles play in similarity?

Solution

1. Because both rhombuses have side length \(5\,\text{cm}\), every pair of corresponding sides has ratio \(1\). Thus, the corresponding sides are proportional. 2. Similarity also requires congruent corresponding angles. 3. A rhombus is not determined by its side length alone. For example, one rhombus could have angles of \(30^\circ\) and \(150^\circ\), while another has angles of \(80^\circ\) and \(100^\circ\). 4. Therefore, equal side lengths do not guarantee similarity. 5. If one angle of the first rhombus is congruent to the corresponding angle of the second, then all corresponding angles are congruent because opposite angles are congruent and adjacent angles are supplementary. This additional condition guarantees similarity.

Answer

a) Two rhombuses can have the same side lengths but different angle measures. Similarity requires congruent corresponding angles as well as proportional corresponding sides. b) A corresponding pair of interior angles must be congruent. Equivalently, the two rhombuses must have the same angle measures.
5148088
A rectangular sign is \(48\,\text{in.}\) wide and \(32\,\text{in.}\) high. It will be enlarged using one of these methods: Method 1: Add \(16\,\text{in.}\) to both the width and the height. Method 2: Multiply both the width and the height by \(1.5\). Calculate the side-length ratios to determine which method produces a sign similar to the original.

Hints

- How do you calculate the ratio of width to height? - What must be true about corresponding side-length ratios in similar rectangles? - Does adding the same amount to both numbers preserve a ratio? - What happens to a ratio when both numbers are multiplied by the same factor?

Solution

1. The original width-to-height ratio is \(\frac{48}{32}=1.5\). 2. With Method 1, the new dimensions are \(48\,\text{in.}+16\,\text{in.}=64\,\text{in.}\) and \(32\,\text{in.}+16\,\text{in.}=48\,\text{in.}\). The new ratio is \(\frac{64}{48}=\frac{4}{3}\approx1.33\). 3. Since \(1.5\ne1.33\), Method 1 does not produce a similar sign. 4. With Method 2, the new dimensions are \(48\,\text{in.}\cdot1.5=72\,\text{in.}\) and \(32\,\text{in.}\cdot1.5=48\,\text{in.}\). The new ratio is \(\frac{72}{48}=1.5\). 5. Method 2 preserves the side-length ratio, and all angles remain \(90^\circ\), so the new sign is similar to the original.

Answer

Only Method 2 produces a sign similar to the original. Method 1 changes the width-to-height ratio from \(1.5\) to approximately \(1.33\), while Method 2 keeps the ratio equal to \(1.5\).
5148358
A copier enlarges a triangular graphic by multiplying every side length by \(1.5\). a) By what percent is the area of the enlarged triangle greater than the original area? b) A student claims, “To double the area of a triangle, I should double every side length.” Check the claim. What factor \(k\) should actually be used to double the area? Round to the nearest hundredth.

Hints

- Area scales with the square of the side-length factor. - Convert the area factor to a percent of the original before finding the percent increase. - Take a square root to find a linear factor from a desired area factor.

Solution

1. For part a), the area factor is \(1.5^2=2.25\). The enlarged area is \(225\%\) of the original, so the increase is \(125\%\). 2. For part b), doubling every side length would multiply the area by \(2^2=4\), so the claim is false. 3. To double the area, \(k^2=2\). Therefore, \(k=\sqrt{2}\approx1.41\).

Answer

a) The area increases by \(125\%\). b) The claim is false. The side lengths should be multiplied by \(k=\sqrt{2}\approx1.41\).
5152038
A landscape architect creates a scale drawing of a triangular property. The drawing has an area of \(50\,\text{cm}^2\), while the actual property has an area of \(450\,\text{m}^2\). a) Find the scale of the drawing. b) A boundary wall on the actual property is \(45\,\text{m}\) long. Find its length on the drawing in centimeters.

Hints

- Convert the two areas to the same unit. - Take the square root of the area ratio to obtain the linear scale. - Convert the actual wall length to centimeters before applying the scale.

Solution

1. Convert the actual area: \(450\,\text{m}^2=4{,}500{,}000\,\text{cm}^2\). 2. The actual-to-drawing area ratio is \(\frac{4500000}{50}=90{,}000\). 3. The actual-to-drawing linear ratio is \(\sqrt{90{,}000}=300\), so the drawing scale is \(1:300\). 4. Convert the wall length: \(45\,\text{m}=4500\,\text{cm}\). 5. Its length on the drawing is \(4500\div300=15\,\text{cm}\).

Answer

a) \(1:300\) b) \(15\,\text{cm}\)
5152488
Triangle \(PQR\) has \(\angle P=55^\circ\) and \(\angle Q=35^\circ\). A dilation with scale factor \(k=3\) maps it to \(\triangle P_1Q_1R_1\). a) Find all three angle measures in \(\triangle P_1Q_1R_1\). b) A student says, “If the side lengths triple, the angle measures must also triple.” Evaluate the claim and explain how dilation affects angle measures.

Hints

- First use the triangle angle-sum theorem. - Which properties remain unchanged under a dilation? - Would tripling a right angle be possible in a triangle?

Solution

1. The third angle in the original triangle is \(\angle R=180^\circ-55^\circ-35^\circ=90^\circ\). 2. A dilation preserves angle measures. 3. Therefore, \(\angle P_1=55^\circ\), \(\angle Q_1=35^\circ\), and \(\angle R_1=90^\circ\). 4. The student’s claim is false. A dilation changes all lengths by the scale factor but preserves the shape and all angle measures.

Answer

a) \(\angle P_1=55^\circ\), \(\angle Q_1=35^\circ\), and \(\angle R_1=90^\circ\) b) The claim is false. Dilations preserve angle measures, regardless of the scale factor.
5153688
An architect makes a scale model of a triangular park at a scale of \(1:50\). The model has an area of \(120\,\text{cm}^2\). a) Find the actual area of the park in square meters. b) One side of the model is \(15\,\text{cm}\) long. Find the actual length of that side in meters.

Hints

- Square the linear scale factor to obtain the area factor. - Convert square centimeters to square meters carefully. - Apply the linear factor directly to the side length.

Solution

1. The linear scale factor from the model to the actual park is \(50\), so the area factor is \(50^2=2500\). 2. The actual area is \(120\cdot2500=300{,}000\,\text{cm}^2\). 3. Since \(10{,}000\,\text{cm}^2=1\,\text{m}^2\), the actual area is \(30\,\text{m}^2\). 4. The actual side length is \(15\cdot50=750\,\text{cm}=7.5\,\text{m}\).

Answer

a) \(30\,\text{m}^2\) b) \(7.5\,\text{m}\)
5153838
A rectangular room is shown on a floor plan at a scale of \(1:50\). On the plan, the room is \(12\,\text{cm}\) long and \(8\,\text{cm}\) wide. a) Find the actual dimensions of the room in meters. b) Find the actual area of the room in square meters. c) An architect makes a new plan on which the room’s area is four times its area on the first plan. What scale is used for the new plan?

Hints

- Use the scale factor to convert plan lengths to actual lengths. - Multiply the actual dimensions to find the area. - An area factor of \(4\) corresponds to what linear factor?

Solution

1. The actual length is \(12\cdot50=600\,\text{cm}=6\,\text{m}\), and the actual width is \(8\cdot50=400\,\text{cm}=4\,\text{m}\). 2. The actual area is \(6\cdot4=24\,\text{m}^2\). 3. If the area on the new plan is multiplied by \(4\), the plan lengths are multiplied by \(\sqrt{4}=2\). 4. Doubling all plan lengths changes the scale from \(1:50\) to \(1:25\).

Answer

a) \(6\,\text{m}\) by \(4\,\text{m}\) b) \(24\,\text{m}^2\) c) \(1:25\)
5153898
A \(10\,\text{cm}\) by \(15\,\text{cm}\) rectangular photo is enlarged proportionally. The enlarged photo has a perimeter of \(1.50\,\text{m}\). a) Find the scale factor \(k\). b) Find the dimensions of the enlarged photo. c) What is the ratio of the original area to the enlarged area?

Hints

- Find the perimeter of the original rectangle. - Convert the enlarged perimeter to centimeters. - Perimeters scale linearly, while areas scale by the square of the linear factor.

Solution

1. The original perimeter is \(2(10+15)=50\,\text{cm}\). 2. The enlarged perimeter is \(1.50\,\text{m}=150\,\text{cm}\). 3. Perimeters scale linearly, so \(k=\frac{150}{50}=3\). 4. The enlarged dimensions are \(10\cdot3=30\,\text{cm}\) and \(15\cdot3=45\,\text{cm}\). 5. The area factor is \(3^2=9\), so the ratio of original area to enlarged area is \(1\) to \(9\).

Answer

a) \(k=3\) b) \(30\,\text{cm}\) by \(45\,\text{cm}\) c) \(1\) to \(9\)
5332008
A rectangular garden bed has vertices \(K(2, 2)\), \(L(6, 2)\), \(M(6, 4)\), and \(N(2, 4)\). Dilate the rectangle from the origin by a scale factor of \(1.5\). a) Find the coordinates of \(K'\), \(L'\), \(M'\), and \(N'\). b) Find the perimeter of each rectangle. How are the perimeters related?
Figure for problem 533200

Hints

- A dilation from the origin multiplies both coordinates of every point by the scale factor. - Find each rectangle's horizontal and vertical side lengths from the coordinates. - Use \(P=2l+2w\) to find each perimeter.

Solution

1. Multiply each coordinate by \(1.5\): \(K(2, 2) \to K'(2 \cdot 1.5, 2 \cdot 1.5)=K'(3, 3)\) \(L(6, 2) \to L'(6 \cdot 1.5, 2 \cdot 1.5)=L'(9, 3)\) \(M(6, 4) \to M'(6 \cdot 1.5, 4 \cdot 1.5)=M'(9, 6)\) \(N(2, 4) \to N'(2 \cdot 1.5, 4 \cdot 1.5)=N'(3, 6)\) 2. The original side lengths are \(4\) units and \(2\) units, so its perimeter is \(2(4+2)=12\) units. 3. The image side lengths are \(6\) units and \(3\) units, so its perimeter is \(2(6+3)=18\) units. 4. Since \(18 \div 12=1.5\), the image perimeter is \(1.5\) times the original perimeter.

Answer

a) \(K'(3, 3)\), \(L'(9, 3)\), \(M'(9, 6)\), and \(N'(3, 6)\) b) Original perimeter: \(12\) units. Image perimeter: \(18\) units. The image perimeter is \(1.5\) times the original perimeter.
5139088
A mathematician claims, “If the circumference of a circle is tripled, then its area is multiplied by \(9\).” a) Use the formulas for circumference \(C\) and area \(A\) to determine whether the claim is always true. b) A circle has an area of \(50\,\text{cm}^2\). Find the area of a new circle whose radius is \(\frac{1}{\sqrt{5}}\) times the original radius.

Hints

- If the circumference is tripled, what happens to the radius? - The radius is squared in the area formula. How does that affect a scale factor? - In part b, compare the new area with the original area without first finding the radius. - What is \((\sqrt{5})^2\)?

Solution

1. The circumference is \(C=2\pi r\). If the circumference is tripled, then the radius must also be tripled, so \(r_{\text{new}}=3r\). 2. The new area is \(A_{\text{new}}=\pi(3r)^2=9\pi r^2=9A\). The claim is true. 3. For part b, \(r_{\text{new}}=\frac{r}{\sqrt{5}}\), so \(A_{\text{new}}=\pi\left(\frac{r}{\sqrt{5}}\right)^2=\frac{1}{5}\pi r^2=\frac{1}{5}A\). 4. Therefore, \(A_{\text{new}}=\frac{1}{5}\cdot50\,\text{cm}^2=10\,\text{cm}^2\).

Answer

a) The claim is true. Tripling the radius multiplies the area by \(3^2=9\). b) \(10\,\text{cm}^2\)
5141188
The area of circle \(2\) is exactly \(16\) times the area of circle \(1\). a) What is the ratio \(r_1:r_2\) of their radii? b) How many times as large is the circumference of circle \(2\) as the circumference of circle \(1\)? c) If the radius of circle \(1\) is cut in half, what fraction of its original area remains?

Hints

- If areas scale by \(k^2\), how do radii scale? - Do circumference and radius have the same scale factor? - What happens when \(\frac{1}{2}\) is squared?

Solution

1. Since \(A=\pi r^2\), an area ratio of \(A_2:A_1=16:1\) gives \(r_2:r_1=\sqrt{16}:1=4:1\). Therefore, \(r_1:r_2=1:4\). 2. Circumference is proportional to radius, so \(C_2=4C_1\). 3. Halving a radius multiplies the area by \(\left(\frac{1}{2}\right)^2=\frac{1}{4}\).

Answer

a) \(r_1:r_2=1:4\) b) The circumference of circle \(2\) is \(4\) times the circumference of circle \(1\). c) \(\frac{1}{4}\) of the original area remains.
5148388
An architect creates three similar rectangular models of a plaza at different scales. Model A has area \(200\,\text{cm}^2\). Model B has area \(1250\,\text{cm}^2\). The perimeter of model C is twice the perimeter of model B. a) Find the ratio of a side length in model A to the corresponding side length in model B. b) By what factor must the area of model A be multiplied to obtain the area of model C?

Hints

- First compare models A and B using their areas. - A perimeter factor is also a linear scale factor. - Multiply successive linear factors, then square the result for the area factor.

Solution

1. For part a), the area factor from A to B is \(\frac{1250}{200}=6.25\). The linear factor is \(\sqrt{6.25}=2.5\), so the side-length ratio from A to B is \(1\) to \(2.5\). 2. The perimeter of C is twice the perimeter of B, so the linear factor from B to C is \(2\). 3. The total linear factor from A to C is \(2.5\cdot2=5\). 4. Therefore, the area factor from A to C is \(5^2=25\).

Answer

a) \(1\) to \(2.5\) b) Multiply the area of model A by \(25\).
5155768
A map has a scale of \(1\) to \(50{,}000\). A square forest is shown with a side length of \(1.2\,\text{cm}\). a) Find the actual side length of the forest in meters. b) Find the actual area of the forest in hectares. Use \(1\,\text{ha}=10{,}000\,\text{m}^2\). c) What is the ratio of the area on the map to the actual area?

Hints

- Interpret what the map scale means for lengths. - Find the area of the actual square. - Area scales by the square of the linear factor. - Keep track of the unit conversions from centimeters to meters and from square meters to hectares.

Solution

1. The actual side length is \(1.2\cdot50{,}000=60{,}000\,\text{cm}=600\,\text{m}\). 2. The actual area is \(600^2=360{,}000\,\text{m}^2\). 3. Convert to hectares: \(360{,}000\div10{,}000=36\,\text{ha}\). 4. The linear ratio from the map to reality is \(1\) to \(50{,}000\), so the area ratio is \(1\) to \(50{,}000^2\), or \(1\) to \(2{,}500{,}000{,}000\).

Answer

a) \(600\,\text{m}\) b) \(36\,\text{ha}\) c) \(1\) to \(2{,}500{,}000{,}000\)

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