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5331655
Each small shape is centered on a grid point. Which geometric shape is centered at \((3, 2)\)?
Figure for problem 533165

Hints

- Read the x-coordinate first. - Then move to the y-coordinate. - Identify the shape centered at that grid location.

Solution

1. Locate x-coordinate \(3\) on the horizontal axis. 2. Move to y-coordinate \(2\). 3. The shape centered at \((3, 2)\) is a triangle.

Answer

The triangle.
5331695
Each small shape is centered on a grid point. What are the coordinates of the center of the pentagon?
Figure for problem 533169

Hints

- Find the shape with five sides. - Read the horizontal coordinate first and the vertical coordinate second.

Solution

1. Identify the shape with five sides. 2. Its center aligns with x-coordinate \(7\) and y-coordinate \(2\). 3. Therefore, the center is at \((7, 2)\).

Answer

\((7, 2)\)
5190745
A trail map uses a coordinate grid. The first number is the x-coordinate, and the second number is the y-coordinate. The Old Oak is at \(E(2, 3)\), the Rock Cave is at \(H(7, 1)\), and the Camp is at \(L(5, 6)\). a) A hiker starts at the Old Oak, moves \(4\) units right, and then moves \(2\) units up. At what coordinates does the hiker stop? b) A supply box is buried \(3\) units left of the Rock Cave and \(4\) units above it. Find the coordinates of the box, \(S\).

Hints

- Moving right or left changes which coordinate? - Moving up or down changes which coordinate? - Picture the coordinate grid or make a quick sketch.

Solution

1. Starting at \(E(2, 3)\), moving \(4\) units right changes the x-coordinate to \(2 + 4 = 6\). Moving \(2\) units up changes the y-coordinate to \(3 + 2 = 5\). The endpoint is \((6, 5)\). 2. Starting at \(H(7, 1)\), moving \(3\) units left changes the x-coordinate to \(7 - 3 = 4\). Moving \(4\) units up changes the y-coordinate to \(1 + 4 = 5\). Therefore, \(S = (4, 5)\).

Answer

a) \((6, 5)\) b) \(S = (4, 5)\)
5190755
Four friends’ homes are located on a coordinate grid: - Alex: \(A(2, 8)\) - Bea: \(B(5, 3)\) - Carl: \(C(10, 8)\) - Dana: \(D(5, 8)\) Use the coordinates to answer each question. a) Who lives farthest to the right? b) Which friends live on the same horizontal line? c) Who lives exactly \(5\) units north of Bea?

Hints

- Use the first coordinate to compare left and right positions. - Points with the same y-coordinate lie on a horizontal line. - Moving north increases the y-coordinate.

Solution

1. Compare the x-coordinates \(2, 5, 10,\) and \(5\). The greatest is \(10\), so Carl lives farthest to the right. 2. Alex, Carl, and Dana all have y-coordinate \(8\), so their homes lie on the same horizontal line. 3. Moving \(5\) units north from \(B(5, 3)\) gives \((5, 3 + 5) = (5, 8)\), which is Dana’s location.

Answer

a) Carl b) Alex, Carl, and Dana c) Dana
5190855
A surveyor records four landmarks on a coordinate map: \(P_1(4.5, 2)\), \(P_2(12, 6.5)\), \(P_3(7, 9)\), and \(P_4(2, 5.5)\). Larger x-values are farther east, and larger y-values are farther north. Find the coordinates of the point that is: a) farthest east. b) farthest north. c) farthest west. d) farthest south.

Hints

- East and west are determined by x-coordinates. - North and south are determined by y-coordinates. - Compare the decimal values carefully.

Solution

1. The greatest x-coordinate is \(12\), so \(P_2(12, 6.5)\) is farthest east. 2. The greatest y-coordinate is \(9\), so \(P_3(7, 9)\) is farthest north. 3. The least x-coordinate is \(2\), so \(P_4(2, 5.5)\) is farthest west. 4. The least y-coordinate is \(2\), so \(P_1(4.5, 2)\) is farthest south.

Answer

a) \(P_2(12, 6.5)\) b) \(P_3(7, 9)\) c) \(P_4(2, 5.5)\) d) \(P_1(4.5, 2)\)
5321725
A theme park map shows several attractions on a coordinate grid: - \(R\): Ferris wheel - \(A\): roller coaster - \(K\): carousel - \(G\): haunted house - \(S\): swinging ship Use the map to answer each question. a) Which attraction is located at \((5, 2)\)? b) What are the coordinates of the haunted house, \(G\)?
Figure for problem 532172

Hints

- Read the x-coordinate before the y-coordinate. - The first coordinate gives the horizontal position. - The second coordinate gives the vertical position.

Solution

1. The point at \((5, 2)\) is labeled \(A\), so it represents the roller coaster. 2. Point \(G\) is \(3\) units right and \(1\) unit up from the origin, so its coordinates are \((3, 1)\).

Answer

a) The roller coaster, point \(A\) b) \((3, 1)\)
5321765
Five points, \(A\), \(B\), \(C\), \(D\), and \(E\), are shown on a coordinate grid. a) What are the coordinates of point \(A\)? b) Which point is located at \((4, 2)\)? c) Start at point \(B\). Move \(2\) units left and \(3\) units up. Which labeled point do you reach?
Figure for problem 532176

Hints

- Read the x-coordinate first and the y-coordinate second. - Moving left decreases x. - Moving up increases y.

Solution

1. Point \(A\) is at x-coordinate \(1\) and y-coordinate \(4\), so \(A = (1, 4)\). 2. The point at \((4, 2)\) is \(B\). 3. Starting at \(B(4, 2)\), moving \(2\) units left and \(3\) units up gives \((4 - 2, 2 + 3) = (2, 5)\), which is point \(D\).

Answer

a) \((1, 4)\) b) \(B\) c) \(D\)
5331455
The graph shows points \(A(-2, -2)\), \(B(2, -2)\), \(C(2, 1)\), \(D(0, 3)\), and \(E(-2, 1)\) connected in alphabetical order, with \(E\) connected back to \(A\). a) What familiar figure does the outline resemble? b) What geometric name describes the outline? c) What are the coordinates of the highest vertex?
Figure for problem 533145

Hints

- Look at the overall outline formed by the connected points. - Count the sides of the closed figure. - The highest point has the greatest y-coordinate.

Solution

1. The horizontal base, two vertical walls, and two slanted roof edges make the outline resemble a house. 2. The closed outline has five sides, so it is a pentagon. 3. The highest vertex is \(D\), which has coordinates \((0, 3)\).

Answer

a) A house b) A pentagon c) \((0, 3)\)
5331465
The graph shows the numbered points connected in order from \(1\) through \(10\), with point \(10\) connected back to point \(1\). What symbol appears? Also write the coordinates of points \(3\), \(6\), and \(9\).
Figure for problem 533146

Hints

- Look at the outer tips of the connected outline. - Read the x-coordinate before the y-coordinate. - Use the labels to locate points \(3\), \(6\), and \(9\).

Solution

1. Reading the graph gives point \(3\) at \((4, 2)\), point \(6\) at \((0, -1)\), and point \(9\) at \((-4, 2)\). 2. The connected points form the outline of a five-pointed star.

Answer

The symbol is a five-pointed star. Point \(3\) is \((4, 2)\), point \(6\) is \((0, -1)\), and point \(9\) is \((-4, 2)\).
5122775
The points \(A(2, 1)\), \(B(4, 3)\), \(C(6, 3)\), and \(D(10, 5)\) are in the coordinate plane. For each point, let the x-coordinate be the denominator and the y-coordinate be the numerator of a fraction. Which points represent equivalent fractions? Explain using equivalent-fraction reasoning.

Hints

- Write the fraction \(\frac{y}{x}\) for each point \((x, y)\). - Simplify each fraction. - Equivalent fractions result when the numerator and denominator are multiplied by the same factor.

Solution

1. The points represent \(A:\frac{1}{2}\), \(B:\frac{3}{4}\), \(C:\frac{3}{6}\), and \(D:\frac{5}{10}\). 2. Simplify: \(\frac{3}{6}=\frac{1}{2}\) and \(\frac{5}{10}=\frac{1}{2}\). 3. Therefore, points \(A\), \(C\), and \(D\) represent equivalent fractions. Point \(C\) is obtained by multiplying both coordinates of \(A\) by \(3\), and point \(D\) by multiplying both coordinates by \(5\).

Answer

Points \(A\), \(C\), and \(D\) represent the same value, \(\frac{1}{2}\).
5122795
The point \(P(3, 2)\) represents the fraction \(\frac{2}{3}\), where the x-coordinate is the denominator and the y-coordinate is the numerator. a) Give the coordinates of the points that represent the fractions obtained by multiplying the numerator and denominator by \(2\), \(3\), and \(4\). b) Describe how to move from one point to the next without recalculating each fraction.

Hints

- Multiply both coordinates by each scale factor. - Compare consecutive x-coordinates and consecutive y-coordinates. - Look for a constant horizontal and vertical change.

Solution

1. Multiplying by \(2\) gives \(\frac{4}{6}\), so the point is \((6, 4)\). 2. Multiplying by \(3\) gives \(\frac{6}{9}\), so the point is \((9, 6)\). 3. Multiplying by \(4\) gives \(\frac{8}{12}\), so the point is \((12, 8)\). 4. From one point to the next, add \((3, 2)\): move \(3\) units right and \(2\) units up.

Answer

a) \((6, 4)\), \((9, 6)\), and \((12, 8)\) b) Move \(3\) units right and \(2\) units up each time.
5349535
The line in the coordinate plane represents all solutions of \(x+y=14\). How many points \((x, y)\) on the line have prime-number values for both \(x\) and \(y\)?
Figure for problem 534953

Hints

- Which numbers less than \(14\) are prime? - Substitute prime values for \(x\) and determine whether the corresponding \(y\) is also prime. - Remember that reversing the coordinates gives a different point unless the coordinates are equal.

Solution

1. Rewrite the equation as \(y=14-x\). 2. List the prime numbers less than \(14\): \(2,3,5,7,11,\) and \(13\). 3. Test which ordered pairs of primes have sum \(14\): \(3+11=14\), \(7+7=14\), and \(11+3=14\). 4. Therefore, the qualifying points are \((3, 11)\), \((7, 7)\), and \((11, 3)\), for a total of \(3\).

Answer

There are \(3\) points: \((3, 11)\), \((7, 7)\), and \((11, 3)\).
5349545
The line shown represents the solutions of \(y-x=2\). How many ordered pairs of prime numbers \((x, y)\) satisfy the equation when both numbers are less than \(20\)?
Figure for problem 534954

Hints

- List the prime numbers less than \(20\). - Which pairs differ by exactly \(2\)? - Locate the corresponding points on the line.

Solution

1. Rewrite the equation as \(y=x+2\). 2. The prime numbers less than \(20\) are \(2,3,5,7,11,13,17,\) and \(19\). 3. Test which ordered prime pairs have a difference of \(2\): \((3, 5)\), \((5, 7)\), \((11, 13)\), and \((17, 19)\). 4. Therefore, there are \(4\) qualifying ordered pairs.

Answer

There are \(4\) pairs: \((3, 5)\), \((5, 7)\), \((11, 13)\), and \((17, 19)\).
5122785
A point has coordinates \((12, 15)\). The x-coordinate is the denominator and the y-coordinate is the numerator of a fraction. a) Which point on the segment from \((0, 0)\) to \((12, 15)\) represents the fraction in simplest form? b) List all points with integer coordinates in the interior of the segment, excluding both endpoints.

Hints

- Simplify the fraction \(\frac{15}{12}\). - Points on the segment have coordinates in the same ratio. - Generate integer multiples of the point representing the simplest fraction.

Solution

1. The point \((12, 15)\) represents \(\frac{15}{12}\). 2. Since the greatest common factor of \(15\) and \(12\) is \(3\), \(\frac{15}{12}=\frac{5}{4}\). The corresponding point is \((4, 5)\). 3. Integer-coordinate points on the segment are multiples of \((4, 5)\): \((0, 0)\), \((4, 5)\), \((8, 10)\), and \((12, 15)\). 4. Excluding the endpoints, the interior points are \((4, 5)\) and \((8, 10)\).

Answer

a) \((4, 5)\) b) \((4, 5)\) and \((8, 10)\)

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