Coordinate Graphing

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Answer Key & Explanations

Coordinate Graphing · Grade 5 · Worksheet 2

1. Hana plots a rectangle on a coordinate grid. The vertices are at (2, 2), (2, 8), (10, 8), and (10, 2). What is the area of the rectangle in square units? Answer: 48 Solution: Identify the side lengths. The points (2, 2) and (10, 2) have the same y-coordinate, so the horizontal side length is 10 - 2 = 8 units.
   Full step-by-step solution

   Step 1: Identify the side lengths. The points (2, 2) and (10, 2) have the same y-coordinate, so the horizontal side length is 10 - 2 = 8 units. The points (2, 2) and (2, 8) have the same x-coordinate, so the vertical side length is 8 - 2 = 6 units. Step 2: Use the area formula for a rectangle: Area = length × width. Step 3: Calculate: Area = 8 × 6 = 48. The area of the rectangle is 48 square units.

2. Plot the points (2, 4), (6, 4), (6, 9), and (2, 9) on a coordinate plane. What shape is formed and what is its area? Answer: rectangle, 20 Solution: Plot the points (2, 4), (6, 4), (6, 9), and (2, 9). The points form a rectangle because the x-coordinates of the bottom points are 2 and 6, and the x-coordinates of the top points are also 2 and 6.
   Full step-by-step solution

   Step 1: Plot the points (2, 4), (6, 4), (6, 9), and (2, 9). Step 2: The points form a rectangle because the x-coordinates of the bottom points are 2 and 6, and the x-coordinates of the top points are also 2 and 6. Similarly, the y-coordinates of the left points are 4 and 9, and the y-coordinates of the right points are also 4 and 9. This means opposite sides are parallel and equal in length. Step 3: Find the length of the base. The base is the horizontal distance between x = 2 and x = 6. Length = 6 - 2 = 4 units. Step 4: Find the height. The height is the vertical distance between y = 4 and y = 9. Height = 9 - 4 = 5 units. Step 5: Calculate the area of the rectangle. Area = length × height = 4 × 5 = 20 square units. The shape is a rectangle with an area of 20 square units.

3. Mason is helping his teacher create a map of the school's new rectangular garden on a coordinate plane. He plots the four corners of the garden at points A(2, 7), B(17, 7), C(17, 12), and D(2, 12). Each unit on the grid represents 1 yard. What is the perimeter of Mason's garden in yards? Answer: 40 yards Solution: Find the length of the bottom side from A(2,7) to B(17,7). Since y-coordinates are the same (7), subtract the x-coordinates: 17 - 2 = 15 yards. The top side from D(2,12) to C(17,12) has the same length: 15 yards.
   Full step-by-step solution

   Step 1: Find the length of the bottom side from A(2,7) to B(17,7). Since y-coordinates are the same (7), subtract the x-coordinates: 17 - 2 = 15 yards. Step 2: The top side from D(2,12) to C(17,12) has the same length: 15 yards. Step 3: Find the width of the left side from A(2,7) to D(2,12). Since x-coordinates are the same (2), subtract the y-coordinates: 12 - 7 = 5 yards. Step 4: The right side from B(17,7) to C(17,12) has the same width: 5 yards. Step 5: Calculate the perimeter: 15 + 5 + 15 + 5 = 40 yards. The perimeter of Mason's garden is 40 yards.

4. A rectangle is drawn on a coordinate plane with vertices at (2.5, 4.75), (8.5, 4.75), (8.5, 1.25), and (2.5, 1.25). What is the area of this rectangle in square units? Answer: 21 Solution: Identify the coordinates of the rectangle's vertices. A = (2.5, 4.75) B = (8.5, 4.75) C = (8.5, 1.25) D = (2.5, 1.25) Determine the length of the horizontal side.
   Full step-by-step solution

   Step 1: Identify the coordinates of the rectangle's vertices. The vertices are: A = (2.5, 4.75) B = (8.5, 4.75) C = (8.5, 1.25) D = (2.5, 1.25) Step 2: Determine the length of the horizontal side. Points A and B have the same y-coordinate (4.75), so they form a horizontal side. Length = difference in x-coordinates = 8.5 - 2.5 = 6.0 units. Step 3: Determine the length of the vertical side. Points A and D have the same x-coordinate (2.5), so they form a vertical side. Height = difference in y-coordinates = 4.75 - 1.25 = 3.5 units. Step 4: Calculate the area of the rectangle. Area = length × height = 6.0 × 3.5. 6.0 × 3.5 = 6 × 3 + 6 × 0.5 = 18 + 3 = 21. Step 5: State the final answer. The area of the rectangle is 21 square units.

5. Plot the points (6, 1), (6, 11), (16, 11), and (16, 1) on a coordinate plane. What shape is formed and what is its area? Answer: rectangle, 100 Solution: Identify the shape. The points (6,1), (6,11), (16,11), and (16,1) form a rectangle because opposite sides are parallel and all angles are right angles. Find the horizontal side length.
   Full step-by-step solution

   Step 1: Identify the shape. The points (6,1), (6,11), (16,11), and (16,1) form a rectangle because opposite sides are parallel and all angles are right angles. Step 2: Find the horizontal side length. The x-coordinates change from 6 to 16, so length = 16 - 6 = 10 units. Step 3: Find the vertical side length. The y-coordinates change from 1 to 11, so width = 11 - 1 = 10 units. Step 4: Calculate the area of the rectangle. Area = length × width = 10 × 10 = 100 square units. The shape is a rectangle with an area of 100 square units.

6. Emma plots a right triangle on a coordinate grid. The vertices are at (3, 5), (3, 13), and (9, 5). What is the area of this triangle in square units? Answer: 24 Solution: Identify the vertices: (3, 5), (3, 13), and (9, 5). Find the vertical leg. Points (3, 5) and (3, 13) have the same x-coordinate (3), so the distance is the difference in y-coordinates: 13 - 5 = 8 units.
   Full step-by-step solution

   Step 1: Identify the vertices: (3, 5), (3, 13), and (9, 5). Step 2: Find the vertical leg. Points (3, 5) and (3, 13) have the same x-coordinate (3), so the distance is the difference in y-coordinates: 13 - 5 = 8 units. Step 3: Find the horizontal leg. Points (3, 5) and (9, 5) have the same y-coordinate (5), so the distance is the difference in x-coordinates: 9 - 3 = 6 units. Step 4: For a right triangle, the legs are the base and height. Area = 1/2 × base × height. Step 5: Calculate area = 1/2 × 6 × 8 = 1/2 × 48 = 24. The area of the triangle is 24 square units.

7. A triangle is drawn on a coordinate plane with vertices at (1, 2), (7, 2), and (4, 8). What is the area of this triangle in square units? Answer: 18 Solution: Identify the base of the triangle. The points (1, 2) and (7, 2) have the same y-coordinate, so this is a horizontal line segment that can serve as the base.
   Full step-by-step solution

   Step 1: Identify the base of the triangle. The points (1, 2) and (7, 2) have the same y-coordinate, so this is a horizontal line segment that can serve as the base. Step 2: Calculate the length of the base: 7 - 1 = 6 units Step 3: The height is the vertical distance from the third vertex (4, 8) to the base. Since the base is at y = 2, the height is 8 - 2 = 6 units Step 4: Apply the triangle area formula: Area = (1/2) × base × height Step 5: Calculate: Area = (1/2) × 6 × 6 = (1/2) × 36 = 18 Step 6: The area of the triangle is 18 square units.

8. Hana draws a parallelogram on a coordinate grid. The vertices are at (10, 15), (21, 15), (16, 21), and (5, 21). What is the area of this parallelogram in square units? Answer: 66 Solution: Identify the base. The top and bottom sides are horizontal. The bottom vertices are (10,15) and (21,15).
   Full step-by-step solution

   Step 1: Identify the base. The top and bottom sides are horizontal. The bottom vertices are (10,15) and (21,15). The length of the base is the distance between these points: 21 - 10 = 11 units. Step 2: Identify the height. The height is the vertical distance between the top and bottom sides. The bottom side is at y = 15 and the top side is at y = 21. The height is 21 - 15 = 6 units. Step 3: Use the area formula for a parallelogram: Area = base x height. Step 4: Calculate: Area = 11 x 6 = 66. The area of the parallelogram is 66 square units.

9. Liam is designing a garden in the shape of a rectangle on a coordinate plane. He plots three corners at (2, 3), (8, 3), and (2, 7). What are the coordinates of the fourth corner of his rectangular garden? Answer: (8, 7) Solution: A = (2, 3) B = (8, 3) C = (2, 7) We need the fourth corner D. In a rectangle, opposite sides are parallel and equal in length, and all angles are 90 degrees.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We are given three corners of a rectangle: A = (2, 3) B = (8, 3) C = (2, 7) We need the fourth corner D. --- **Step 2: Identify sides of the rectangle** In a rectangle, opposite sides are parallel and equal in length, and all angles are 90 degrees. Let's plot them mentally: - A and B have the same y-coordinate (y = 3), so AB is a horizontal line from (2, 3) to (8, 3). Length of AB = 8 - 2 = 6 units. - A and C have the same x-coordinate (x = 2), so AC is a vertical line from (2, 3) to (2, 7). Length of AC = 7 - 3 = 4 units. So A is the bottom-left corner, B is bottom-right, C is top-left. --- **Step 3: Find the fourth corner** We know: - B = (8, 3) is bottom-right. - C = (2, 7) is top-left. The fourth corner D will be top-right. From B: go vertically up the same length as AC (4 units) → y = 3 + 4 = 7. From C: go horizontally right the same length as AB (6 units) → x = 2 + 6 = 8. So D = (8, 7). --- **Step 4: Check rectangle properties** Coordinates: A = (2, 3), B = (8, 3), C = (2, 7), D = (8, 7) AB horizontal length = 6 CD horizontal length = 8 - 2 = 6 ✓ AC vertical length = 4 BD vertical length = 7 - 3 = 4 ✓ All sides parallel to axes, opposite sides equal. --- **Final Answer:** (8, 7)