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Scale drawings

Learning goals

  • Interprets the scale factor relating drawing and actual dimensions.
  • Computes unknown lengths using proportional reasoning.
  • Converts units when drawing and actual measurements use different units.
  • Checks that all dimensions use the same scale.

Click problems to add them to your worksheet.

Scale drawings 1
A small kitchen table is actually \(150\,\text{cm}\) long. In a scale drawing, it is \(15\,\text{cm}\) long. What scale was used?

Hints

- Compare the actual length with the drawing length. - Determine how many times the drawing length fits into the actual length. - Write the drawing-to-actual relationship as a scale.

Solution

1. Compare the actual length with the drawing length: \(150\,\text{cm} \div 15\,\text{cm} = 10\). 2. The actual table is \(10\) times as long as the drawing, so the scale is \(1{:}10\).

Answer

\(1{:}10\)
Scale drawings 2
On a scale drawing, a wall segment that is \(2.5\,\text{inches}\) long represents an actual wall that is \(15\,\text{feet}\) long. State the scale in the form “\(1\,\text{inch}\) represents ___ feet,” and explain how you determined it.

Hints

- Compare the actual length with the corresponding drawing length. - The requested scale asks for the actual feet represented by one drawing inch. - Think about which length you should divide by the other.

Solution

1. Divide actual length by drawing length: \(15\div2.5=6\). 2. Therefore, each \(1\,\text{inch}\) on the drawing represents \(6\,\text{feet}\) in the room.

Answer

\(1\,\text{inch}\) represents \(6\,\text{feet}\), since \(15\div2.5=6\) feet per drawing inch.
Scale drawings 3
A floor plan for Cedar Grove Library uses the scale \(1\,\text{inch}\) represents \(5\,\text{feet}\). A wall measures \(3.5\,\text{inches}\) on the plan. How long is the actual wall?

Hints

- Decide whether the actual wall should be longer or shorter than its drawing length. - Use the number of actual feet represented by each drawing inch. - Apply the scale to the \(3.5\)-inch drawing length.

Solution

1. Each drawing inch represents \(5\,\text{ft}\). 2. Compute \(3.5\cdot5=17.5\).

Answer

The actual wall is \(17.5\,\text{ft}\) long.
Scale drawings 4
A trail map uses the scale \(1\,\text{cm}\) represents \(4\,\text{km}\). A trail segment is actually \(18\,\text{km}\) long. How long should that segment be on the map?

Hints

- This time the actual length is known and the drawing length is unknown. - Ask how many groups of \(4\,\text{km}\) fit into \(18\,\text{km}\). - The answer should be a length on the map, measured in centimeters.

Solution

1. Each \(1\,\text{cm}\) on the map represents \(4\,\text{km}\) in reality. 2. Compute \(18\div4=4.5\).

Answer

The segment should be \(4.5\,\text{cm}\) long on the map.
Scale drawings 5
Each space between neighboring pegs is one grid unit. Are figures a) and b) scale copies? If so, find the scale factor from a) to b) by comparing their horizontal and vertical sides.
Figure for problem 559508

Hints

- Count spaces along a side, not individual pegs. - Compare a horizontal side first. - Check that the vertical side changes by the same factor.

Solution

1. Figure a) is \(2\) grid units wide and \(1\) unit tall. 2. Figure b) is \(4\) grid units wide and \(2\) units tall. 3. Both side-length ratios from a) to b) equal \(2\), so b) is a scale copy with factor \(2\).

Answer

Yes. The horizontal factor is \(4\div2=2\), and the vertical factor is \(2\div1=2\). The scale factor from a) to b) is \(2\).
Scale drawings 6
Each space between neighboring pegs represents \(2\,\text{feet}\) along the edge of a garden. What is the actual length of the garden's shorter side? Explain how the diagram gives that length.
Figure for problem 559509

Hints

- Count the spaces along the shorter edge. - Remember the diagram gives the distance for one space. - Multiply spaces by feet per space.

Solution

1. The shorter side spans \(3\) grid spaces. 2. Each space represents \(2\) feet, so the shorter side measures \(3\cdot2=6\) feet.

Answer

\(6\,\text{ft}\), because the shorter side spans \(3\) spaces of \(2\) feet each.
Scale drawings 7
A photograph shows an ant at \(12\) times its actual size. The ant measures \(5.4\,\text{cm}\) in the photograph. What is the ant's actual length in millimeters?

Hints

- A magnification of \(12\) means the image length is \(12\) times the actual length. - Decide whether the actual ant should be longer or shorter than the image. - Convert centimeters to millimeters after finding the actual length.

Solution

1. Divide the image length by the scale factor: \(5.4\div12=0.45\,\text{cm}\). 2. Convert centimeters to millimeters: \(0.45\,\text{cm}=4.5\,\text{mm}\).

Answer

The ant is \(4.5\,\text{mm}\) long.
Scale drawings 8
Each space between neighboring pegs is one grid unit. Nora says figure b) is bigger than figure a), so it must be a scale copy. Is she right? Show a horizontal comparison and a vertical comparison.
Figure for problem 559510

Hints

- A larger figure is not always a scale copy. - Count the side lengths in grid spaces for each figure. - Check whether the two directions use the same multiplier.

Solution

1. Figure a) is \(2\) units wide and \(3\) units tall. Figure b) is \(4\) units wide and \(5\) units tall. 2. The horizontal factor is \(4\div2=2\). 3. The vertical factor is \(5\div3=\frac{5}{3}\), not \(2\). 4. The sides do not scale by the same factor, so the figures are not scale copies.

Answer

No. The horizontal factor is \(2\), while the vertical factor is \(\frac{5}{3}\). The factors must match for a scale copy.
Scale drawings 9
The two posters are scale copies. Find the height of figure b) in inches. Explain how the corresponding widths determine the scale factor.
Figure for problem 559511

Hints

- Use two corresponding sides whose lengths are labeled. - Find the multiplier from the smaller poster to the larger poster. - Apply that multiplier to the shorter poster's height.

Solution

1. The widths are \(3\) inches and \(9\) inches, so the scale factor from a) to b) is \(9\div3=3\). 2. The height of a) is \(2\) inches. Multiply by the same factor: \(2\cdot3=6\) inches.

Answer

\(6\,\text{in}\). The width grows from \(3\) to \(9\) inches, a factor of \(3\), so the height grows from \(2\) to \(6\) inches.
Scale drawings 10
The drawing shows a rectangular patio. The long side of the real patio is \(10\,\text{m}\). a) How many meters does \(1\,\text{cm}\) on the drawing represent? b) Suppose a path would be \(1.5\,\text{cm}\) long on the drawing. How long would it be in real life?
Figure for problem 559512

Hints

- What length on the drawing matches the real patio's long side? - Divide actual length in meters by drawing length in centimeters. - Apply the same real-world meters-per-centimeter rate to the path.

Solution

1. The drawing's long side is \(4\) centimeters and represents \(10\) meters. 2. Each drawing centimeter represents \(10\div4=2.5\) meters. 3. A drawing length of \(1.5\) centimeters represents \(1.5\cdot2.5=3.75\) meters.

Answer

a) \(1\,\text{cm}\) on the drawing represents \(2.5\,\text{m}\) in real life. b) \(3.75\,\text{m}\).
Scale drawings 11
A microorganism is actually \(0.25\,\text{mm}\) long. a) How long does it appear under \(40\)-times magnification? Give the answer in centimeters. b) Arjun wants the image to be exactly \(2\,\text{cm}\) long. What magnification factor is needed?

Hints

- Relate actual length, image length, and magnification factor. - Use the same length unit before forming a scale factor. - For part b), compare the desired image length with the actual length.

Solution

1. For a), multiply the actual length by the scale factor: \(0.25\cdot40=10\,\text{mm}\). 2. Convert \(10\,\text{mm}\) to \(1\,\text{cm}\). 3. For b), convert the desired image length: \(2\,\text{cm}=20\,\text{mm}\). 4. Divide image length by actual length: \(20\div0.25=80\).

Answer

a) \(1\,\text{cm}\) b) \(80\)-times magnification
Scale drawings 12
A technical part is only \(0.08\,\text{mm}\) thick. In a design drawing, it is shown as \(4\,\text{mm}\) thick. a) Find the scale factor of the drawing. b) A second part is \(0.12\,\text{mm}\) thick. How thick will it appear in the same drawing? c) If the first part is enlarged by a factor of \(1000\), how thick will it appear in centimeters?

Hints

- Compare the drawn thickness with the actual thickness to find the scale factor. - Apply the same scale factor to the second part. - Convert millimeters to centimeters after applying the factor in part c).

Solution

1. For a), divide drawing thickness by actual thickness: \(4\div0.08=50\). The scale factor is \(50\). 2. For b), apply the same factor: \(0.12\cdot50=6\,\text{mm}\). 3. For c), \(0.08\cdot1000=80\,\text{mm}=8\,\text{cm}\).

Answer

a) \(50\) b) \(6\,\text{mm}\) c) \(8\,\text{cm}\)
Scale drawings 13
On a number line drawing, \(1\,\text{cm}\) represents \(0.2\) unit. a) Describe where to place \(-1.4\), \(0.6\), \(-0.8\), and \(1.2\) relative to \(0\). b) Find the distance between the least and greatest of the four numbers. Give the numerical distance and the corresponding length on the drawing.

Hints

- Identify the least and greatest numbers. - Find their distance by subtracting the lesser value from the greater value. - Divide by \(0.2\) to convert a numerical distance to centimeters on the drawing.

Solution

1. Since \(1\,\text{cm}\) represents \(0.2\), divide each absolute value by \(0.2\). Thus, \(-1.4\) is \(7\,\text{cm}\) left of \(0\), \(0.6\) is \(3\,\text{cm}\) right, \(-0.8\) is \(4\,\text{cm}\) left, and \(1.2\) is \(6\,\text{cm}\) right. 2. The least number is \(-1.4\), and the greatest is \(1.2\). 3. Their numerical distance is \(1.2-(-1.4)=2.6\). 4. On the drawing, the length is \(2.6\div0.2=13\,\text{cm}\).

Answer

a) \(-1.4\): \(7\,\text{cm}\) left; \(0.6\): \(3\,\text{cm}\) right; \(-0.8\): \(4\,\text{cm}\) left; \(1.2\): \(6\,\text{cm}\) right. b) The numerical distance is \(2.6\), which is \(13\,\text{cm}\) on the drawing.
Scale drawings 14
Three maps of a city use these scales: Map A: \(1:2000\) Map B: \(1:25{,}000\) Map C: \(1:250{,}000\) a) On which map does \(1\,\text{cm}\) represent the greatest actual distance? b) Which map is best for seeing details such as individual buildings and street names? Explain. c) How many meters does \(1\,\text{cm}\) represent on Map B?

Hints

- Compare the numbers after the colon. - A larger scale denominator means a larger actual area is compressed onto the map. - Convert centimeters to meters for c).

Solution

1. A larger denominator means a greater reduction. Therefore, \(1\,\text{cm}\) represents the greatest actual distance on Map C. 2. A smaller denominator shows objects at a larger size and with more detail. Therefore, Map A is best for individual buildings and street names. 3. On Map B, \(1\,\text{cm}\) represents \(25{,}000\,\text{cm}\). Convert to meters: \(25{,}000\div 100=250\,\text{m}\).

Answer

a) Map C b) Map A, because it has the least reduction and shows the most detail. c) \(250\,\text{m}\)
Scale drawings 15
A school campus is \(200\,\text{m}\) long. It will be drawn using two scales. Drawing A uses \(1:1000\). Drawing B uses \(1:2500\). a) Find the campus length in each drawing, in centimeters. b) Which drawing shows the campus as a smaller image? Explain without doing another calculation.

Hints

- Convert the actual length to centimeters. - Divide by each scale denominator. - A larger denominator creates a smaller drawing.

Solution

1. Convert the actual length: \(200\,\text{m}=20{,}000\,\text{cm}\). 2. Drawing A has length \(20{,}000\div 1000=20\,\text{cm}\). 3. Drawing B has length \(20{,}000\div 2500=8\,\text{cm}\). 4. Drawing B is smaller because a larger scale denominator means a greater reduction.

Answer

a) Drawing A: \(20\,\text{cm}\) Drawing B: \(8\,\text{cm}\) b) Drawing B, because \(1:2500\) reduces the actual length more than \(1:1000\).
Scale drawings 16
The figure is a scale drawing of a rectangular reading nook. The scale is \(1\,\text{inch}\) in the drawing for every \(2.5\,\text{feet}\) in the actual room. a) Find the actual length and width. b) Find the actual floor area.
Figure for problem 550217

Hints

- Use the labeled drawing dimensions rather than measuring the image. - Apply the same scale relationship to each length. - Find the actual area only after converting both dimensions to actual lengths.

Solution

1. The drawing measures \(4\,\text{in}\times3\,\text{in}\). 2. The actual dimensions are \(4\cdot2.5=10\,\text{ft}\) and \(3\cdot2.5=7.5\,\text{ft}\). 3. The actual floor area is \(10\cdot7.5=75\,\text{ft}^2\).

Answer

a) \(10\,\text{ft}\times7.5\,\text{ft}\) b) \(75\,\text{ft}^2\)
Scale drawings 17
Each grid step represents \(1\,\text{drawing unit}\). Figure a) is the original drawing. Which of figures b) or c) is a scale copy of figure a)? Give the scale factor from a) to the matching figure and explain your choice.
Figure for problem 550218

Hints

- Compare the horizontal and vertical lengths of corresponding triangles. - A scale copy has the same length multiplier in every direction. - Check more than one pair of corresponding lengths before deciding.

Solution

1. In a), the triangle has a base of \(2\) units and a height of \(2\) units. 2. In b), the corresponding base and height are both \(4\) units, so each is twice its length in a). 3. In c), the base is \(4\) units but the height is \(3\) units. Those give different factors, \(2\) and \(1.5\). 4. Only b) preserves the same multiplier for both corresponding dimensions, so its scale factor is \(2\).

Answer

Figure b) is the scale copy, with factor \(2\). Its base and height both change from \(2\) units to \(4\) units. Figure c) has a \(4\)-unit base but only a \(3\)-unit height, so it is not a scale copy.
Scale drawings 18
The figure is a \(1{:}6\) scale drawing of a rectangular sign. A second drawing of the same sign will use a \(1{:}4\) scale. a) What dimensions should the second drawing have? b) By what factor are the side lengths in the second drawing larger than the corresponding side lengths in the first drawing?
Figure for problem 550220

Hints

- Use the first scale to determine the actual dimensions represented by the figure. - Then apply the new scale to those same actual dimensions. - Compare corresponding side lengths from the two drawings to find the enlargement factor.

Solution

1. The first drawing measures \(6\,\text{in}\times4\,\text{in}\), so the actual sign measures \(36\,\text{in}\times24\,\text{in}\). 2. At a \(1{:}4\) scale, the second drawing measures \(36 \div 4=9\,\text{in}\) by \(24 \div 4=6\,\text{in}\). 3. The side-length factor from the first drawing to the second is \(9 \div 6=1.5\), which also matches \(6 \div 4=1.5\).

Answer

a) \(9\,\text{in}\times6\,\text{in}\) b) The side lengths are multiplied by \(1.5\).
Scale drawings 19
The geoboard shows a scale drawing of a banner shape. Treat the lower-left vertex of the shown polygon as \((0,0)\). One peg spacing is \(1\) unit, positive x is to the right, and positive y is upward. A second drawing must reproduce the same shape at a scale factor of \(2\), with its corresponding lower-left vertex also at \((0,0)\). a) Starting at the lower-left vertex and moving counterclockwise, list the coordinates of the vertices in the shown drawing. b) List the coordinates of the corresponding vertices in the second drawing. c) Explain how the scale factor changes every side length.
Figure for problem 554539

Hints

- Use peg spacings, not physical measurements of the displayed image. - Record each vertex relative to the stated lower-left origin before changing the scale. - A scale factor changes coordinate displacements from the origin by the same multiplicative factor.

Solution

1. Counting peg spacings from the lower-left vertex, the shown vertices are \((0,0)\), \((3,0)\), \((3,2)\), \((1,3)\), and \((0,2)\). 2. A scale factor of \(2\) doubles each coordinate measured from the common origin. 3. The second drawing therefore has vertices \((0,0)\), \((6,0)\), \((6,4)\), \((2,6)\), and \((0,4)\). 4. Because every coordinate displacement is doubled, every side length is doubled and the shape is reproduced at the required scale.

Answer

a) \((0,0)\), \((3,0)\), \((3,2)\), \((1,3)\), \((0,2)\) b) \((0,0)\), \((6,0)\), \((6,4)\), \((2,6)\), \((0,4)\) c) Every side length is multiplied by \(2\).
Scale drawings 20
The figure is a scale drawing of an L-shaped room. The scale is \(1\,\text{inch}\) in the drawing for every \(4\,\text{feet}\) in the actual room. Find the actual floor area of the room.
Figure for problem 550219

Hints

- Can you break the room into simpler shapes, or compare it with a larger shape? - Which lengths are shown, and which others can you work out from them? - What happens to both dimensions of a rectangle when you convert from drawing inches to actual feet?

Solution

1. View the drawing as a \(5\,\text{in}\times4\,\text{in}\) rectangle with a \(3\,\text{in}\times2\,\text{in}\) rectangle missing. 2. The drawing area is \(5\cdot4-3\cdot2=14\,\text{in}^2\). 3. Each drawing length represents \(4\) actual feet, so the number of square feet is multiplied by \(4^2=16\). 4. The actual floor area is \(14\cdot16=224\,\text{ft}^2\).

Answer

The actual floor area is \(224\,\text{ft}^2\).

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