Coordinate Graphing

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Answer Key & Explanations

Coordinate Graphing · Grade 5 · Worksheet 3

1. 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.

2. 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.

3. Plot the points (12, 15), (12, 23), (28, 23), and (28, 15) on a coordinate plane. What is the area of the shape formed? Answer: 128 Solution: Plot the points (12, 15), (12, 23), (28, 23), and (28, 15). Connect them in order: (12,15) to (12,23) to (28,23) to (28,15) and back to (12,15).
   Full step-by-step solution

   Step 1: Plot the points (12, 15), (12, 23), (28, 23), and (28, 15). Connect them in order: (12,15) to (12,23) to (28,23) to (28,15) and back to (12,15). The shape has all right angles and opposite sides equal, so it is a rectangle. Step 2: Find the length (horizontal distance). The x-coordinates are 12 and 28, so length = 28 - 12 = 16 units. Step 3: Find the width (vertical distance). The y-coordinates are 15 and 23, so width = 23 - 15 = 8 units. Step 4: Calculate the area of the rectangle. Area = length × width = 16 × 8 = 128 square units. The answer is 128.

4. Hana is helping her teacher design a rectangular mural for the school hallway. On a coordinate grid where each unit represents 1 foot, she marks three corners of the mural at points (14, 12), (14, 22), and (30, 12). What are the coordinates of the fourth corner she needs to complete the rectangle, and what is the area of the mural in square feet? Answer: (30, 22) and 160 square feet Solution: Identify the given points: (14, 12), (14, 22), and (30, 12). Notice that (14, 12) and (14, 22) share the same x-coordinate (14), so they form a vertical side. The length of this vertical side is 22 - 12 = 10 feet.
   Full step-by-step solution

   Step 1: Identify the given points: (14, 12), (14, 22), and (30, 12). Notice that (14, 12) and (14, 22) share the same x-coordinate (14), so they form a vertical side. The length of this vertical side is 22 - 12 = 10 feet. Step 2: Notice that (14, 12) and (30, 12) share the same y-coordinate (12), so they form a horizontal side. The length of this horizontal side is 30 - 14 = 16 feet. Step 3: The fourth corner must have the x-coordinate of (30, 12) and the y-coordinate of (14, 22) to complete the rectangle. So the fourth corner is (30, 22). Step 4: Check: The vertical side from (30, 12) to (30, 22) has length 22 - 12 = 10 feet, matching the other vertical side. The horizontal side from (14, 22) to (30, 22) has length 30 - 14 = 16 feet, matching the other horizontal side. Step 5: Area of the rectangle = length × width = 16 feet × 10 feet = 160 square feet. The fourth corner is at (30, 22) and the area of the mural is 160 square feet.

5. (3.5 × 4.2) + (7.8 ÷ 2.6) = ? Answer: 17.7 Solution: We have: (3.5 × 4.2) + (7.8 ÷ 2.6) First, handle the multiplication inside the first parentheses.
   Full step-by-step solution

   Let's solve step-by-step. We have: (3.5 × 4.2) + (7.8 ÷ 2.6) Step 1: First, handle the multiplication inside the first parentheses. 3.5 × 4.2 3.5 × 4 = 14 3.5 × 0.2 = 0.7 Add them: 14 + 0.7 = 14.7 So, (3.5 × 4.2) = 14.7 Step 2: Next, handle the division inside the second parentheses. 7.8 ÷ 2.6 We can think of it as 78 ÷ 26. 26 × 3 = 78, so 78 ÷ 26 = 3. Thus, 7.8 ÷ 2.6 = 3. So, (7.8 ÷ 2.6) = 3 Step 3: Now add the two results. 14.7 + 3 = 17.7 Final answer: 17.7

6. Find the perimeter of a rectangle with vertices at (2, 3), (7, 3), (7, 8), and (2, 8) on the coordinate plane. Answer: 20 Solution: Identify the coordinates of the rectangle. The vertices are: A = (2, 3), B = (7, 3), C = (7, 8), D = (2, 8). Find the length of side AB.
   Full step-by-step solution

   Step 1: Identify the coordinates of the rectangle. The vertices are: A = (2, 3), B = (7, 3), C = (7, 8), D = (2, 8). Step 2: Find the length of side AB. Points A and B have the same y-coordinate (3), so the length is the difference in x-coordinates. Length AB = 7 - 2 = 5. Step 3: Find the length of side BC. Points B and C have the same x-coordinate (7), so the length is the difference in y-coordinates. Length BC = 8 - 3 = 5. Step 4: Check if the rectangle has sides of equal length for opposite sides. Since it is a rectangle, opposite sides are equal. So, length CD = length AB = 5, and length DA = length BC = 5. Step 5: Alternatively, verify with another side for clarity. Points C = (7, 8) and D = (2, 8) have the same y-coordinate (8), so length CD = 7 - 2 = 5. Points D = (2, 8) and A = (2, 3) have the same x-coordinate (2), so length DA = 8 - 3 = 5. Step 6: Calculate the perimeter. Perimeter of a rectangle = 2 * (length + width) = 2 * (5 + 5) = 2 * 10 = 20. Final Answer: 20

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

   Let's find the area of the rectangular garden step by step. Step 1: Identify the coordinates of the rectangle's corners. A(2, 3), B(8, 3), C(8, 7), D(2, 7) Step 2: Find the length of side AB. Points A and B have the same y-coordinate (3), so AB is horizontal. Length AB = difference in x-coordinates = 8 - 2 = 6 units. Step 3: Find the length of side BC. Points B and C have the same x-coordinate (8), so BC is vertical. Length BC = difference in y-coordinates = 7 - 3 = 4 units. Step 4: Confirm the shape is a rectangle. Since AB is horizontal and BC is vertical, angle ABC is 90 degrees. Also, opposite sides are equal: AB = CD = 6, BC = AD = 4. Step 5: Calculate the area of the rectangle. Area = length × width = AB × BC = 6 × 4 = 24 square units. Thus, the area of Lily's garden is 24 square units.

8. Plot the points (7, 2), (7, 17), (22, 17), and (22, 2) on a coordinate plane. What shape is formed and what is its area? Answer: rectangle, 225 Solution: Plot the points (7, 2), (7, 17), (22, 17), and (22, 2). Connect them in order: (7,2) to (7,17) to (22,17) to (22,2) and back to (7,2).
   Full step-by-step solution

   Step 1: Plot the points (7, 2), (7, 17), (22, 17), and (22, 2). Connect them in order: (7,2) to (7,17) to (22,17) to (22,2) and back to (7,2). The shape has four right angles and opposite sides parallel and equal, so it is a rectangle. Step 2: Find the horizontal side length. The x-coordinates are 7 and 22, so length = 22 - 7 = 15 units. Step 3: Find the vertical side length. The y-coordinates are 2 and 17, so width = 17 - 2 = 15 units. Step 4: Since both side lengths are 15 units, the shape is actually a square (a special type of rectangle). Area of a square = side × side = 15 × 15 = 225 square units. The answer is rectangle, 225.

9. 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.