Coordinate Graphing

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Answer Key & Explanations

Coordinate Graphing · Grade 5 · Worksheet 1

1. 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)

2. Sophia plots a triangle on a coordinate plane. The vertices of the triangle are at (7, 8), (7, 14), and (13, 8). What is the area of this triangle in square units? Answer: 18 Solution: Identify the vertices: (7, 8), (7, 14), and (13, 8). Points (7, 8) and (7, 14) have the same x-coordinate (7), so they form a vertical side.
   Full step-by-step solution

   Step 1: Identify the vertices: (7, 8), (7, 14), and (13, 8). Step 2: Points (7, 8) and (7, 14) have the same x-coordinate (7), so they form a vertical side. The length of this vertical side is the difference in y-coordinates: 14 - 8 = 6 units. This is one leg of the right triangle. Step 3: Points (7, 8) and (13, 8) have the same y-coordinate (8), so they form a horizontal side. The length of this horizontal side is the difference in x-coordinates: 13 - 7 = 6 units. This is the other leg of the right triangle. 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 × 6 = 1/2 × 36 = 18. The area of the triangle is 18 square units.

3. Plot the points (4, 1), (4, 9), (12, 9), and (12, 1) on a coordinate plane. What shape is formed and what is its perimeter? Answer: rectangle, 32 Solution: Identify the shape by examining the coordinates. The x-coordinates are 4 and 12, so the horizontal distance is 12 - 4 = 8 units. The y-coordinates are 1 and 9, so the vertical distance is 9 - 1 = 8 units.
   Full step-by-step solution

   Step 1: Identify the shape by examining the coordinates. The x-coordinates are 4 and 12, so the horizontal distance is 12 - 4 = 8 units. The y-coordinates are 1 and 9, so the vertical distance is 9 - 1 = 8 units. Since all four points form right angles and opposite sides are equal, this is a square. Step 2: Calculate the perimeter of the square. Perimeter of a square = 4 × side length Side length = 8 units Perimeter = 4 × 8 = 32 units The answer is rectangle, 32.

4. Olivia is drawing a map of her neighborhood park on a coordinate grid. She marks the four corners of the rectangular picnic area at points A(1, 1), B(1, 9), C(7, 9), and D(7, 1). If each unit on the grid represents 5 feet, what is the perimeter of the picnic area in feet? Answer: 140 feet Solution: Find the width (vertical distance) using points A(1,1) and B(1,9). Since the x-coordinates are the same, subtract the y-coordinates: 9 - 1 = 8 units. Find the length (horizontal distance) using points A(1,1) and D(7,1).
   Full step-by-step solution

   Step 1: Find the width (vertical distance) using points A(1,1) and B(1,9). Since the x-coordinates are the same, subtract the y-coordinates: 9 - 1 = 8 units. Step 2: Find the length (horizontal distance) using points A(1,1) and D(7,1). Since the y-coordinates are the same, subtract the x-coordinates: 7 - 1 = 6 units. Step 3: The perimeter in grid units is 2 × (length + width) = 2 × (6 + 8) = 2 × 14 = 28 units. Step 4: Convert to feet. Each unit represents 5 feet, so multiply the grid perimeter by 5: 28 × 5 = 140 feet. The perimeter of the picnic area is 140 feet.

5. Noah plots a rectangle on a coordinate plane. The vertices are at (1, 1), (1, 6), (6, 6), and (6, 1). What is the area of this rectangle in square units? Answer: 25 Solution: Identify the side lengths. The points (1,1) and (1,6) share the same x-coordinate, so the vertical side length is 6 - 1 = 5 units.
   Full step-by-step solution

   Step 1: Identify the side lengths. The points (1,1) and (1,6) share the same x-coordinate, so the vertical side length is 6 - 1 = 5 units. The points (1,1) and (6,1) share the same y-coordinate, so the horizontal side length is 6 - 1 = 5 units. Step 2: Use the area formula for a rectangle: Area = length x width. Step 3: Calculate: Area = 5 x 5 = 25. The area of the rectangle is 25 square units.

6. Plot the points (3, 2), (8, 2), (8, 6), and (3, 6) on a coordinate plane. What is the area of the shape formed? Answer: 20 Solution: Plot the points (3, 2), (8, 2), (8, 6), and (3, 6). Connect the points in order. The shape is a rectangle.
   Full step-by-step solution

   Step 1: Plot the points (3, 2), (8, 2), (8, 6), and (3, 6). Step 2: Connect the points in order. The shape is a rectangle. Step 3: Find the length by calculating the difference in the x-coordinates: 8 - 3 = 5 units. Step 4: Find the width by calculating the difference in the y-coordinates: 6 - 2 = 4 units. Step 5: Calculate the area using the formula for a rectangle: Area = length × width. Step 6: Area = 5 × 4 = 20 square units. The area is 20.

7. Lily is designing a garden plot on a coordinate plane. She marks four corners for a rectangular flower bed at points A(2, 3), B(8, 3), C(8, 7), and D(2, 7). What is the area of Lily's flower bed in square units? Answer: 24 Solution: Points: A(2, 3), B(8, 3), C(8, 7), D(2, 7). Identify the length and width. Notice that A and B have the same y-coordinate (y = 3), so AB is horizontal.
   Full step-by-step solution

   Let's find the area step by step. Step 1: Understand the coordinates of the rectangle. Points: A(2, 3), B(8, 3), C(8, 7), D(2, 7). Step 2: Identify the length and width. Notice that A and B have the same y-coordinate (y = 3), so AB is horizontal. The distance between A(2, 3) and B(8, 3) is: Length = 8 - 2 = 6 units. Similarly, A and D have the same x-coordinate (x = 2), so AD is vertical. The distance between A(2, 3) and D(2, 7) is: Width = 7 - 3 = 4 units. Step 3: Calculate the area of the rectangle. Area = length × width = 6 × 4 = 24 square units. Step 4: Conclusion. The area of Lily's flower bed is 24 square units.

8. Plot the points (8, 9), (8, 15), (18, 15), and (18, 9) on a coordinate plane. What shape is formed and what is its area? Answer: rectangle, 60 Solution: Identify the shape. The points are (8,9), (8,15), (18,15), and (18,9). The x-coordinates are 8 and 18, and the y-coordinates are 9 and 15.
   Full step-by-step solution

   Step 1: Identify the shape. The points are (8,9), (8,15), (18,15), and (18,9). The x-coordinates are 8 and 18, and the y-coordinates are 9 and 15. Opposite sides are parallel and all angles are right angles, so it is a rectangle. Step 2: Find the length (horizontal side). The x-coordinates change from 8 to 18, so length = 18 - 8 = 10 units. Step 3: Find the width (vertical side). The y-coordinates change from 9 to 15, so width = 15 - 9 = 6 units. Step 4: Calculate the area. Area of a rectangle = length × width = 10 × 6 = 60 square units. The answer is rectangle, 60.

9. Charlotte is designing a rectangular flower bed on a coordinate grid. She plots three of the four corners at points A(8, 12), B(8, 20), and C(22, 20). What are the coordinates of the fourth corner, D, needed to complete the rectangle? Answer: (22, 12) Solution: Identify the given points: A(8, 12), B(8, 20), C(22, 20). Notice that A and B share the same x-coordinate (8), so they form a vertical side.
   Full step-by-step solution

   Step 1: Identify the given points: A(8, 12), B(8, 20), C(22, 20). Step 2: Notice that A and B share the same x-coordinate (8), so they form a vertical side. Step 3: Notice that B and C share the same y-coordinate (20), so they form a horizontal side. Step 4: A rectangle has opposite sides equal and parallel. The fourth corner D must share the same x-coordinate as C (22) and the same y-coordinate as A (12). Step 5: Therefore, D is at (22, 12). The coordinates of the fourth corner are (22, 12).