Coordinate Graphing

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Answer Key & Explanations

Coordinate Graphing · Grade 5 · Worksheet 3

1. Plot the points (4, 1), (9, 1), (9, 7), and (4, 7) on a coordinate plane. What shape is formed and what is its perimeter? Answer: rectangle, 22 Solution: Plot the points (4,1), (9,1), (9,7), and (4,7). Connect the points in order: (4,1) to (9,1) to (9,7) to (4,7) and back to (4,1). The shape has all right angles and opposite sides equal, so it is a rectangle.
   Full step-by-step solution

   Step 1: Plot the points (4,1), (9,1), (9,7), and (4,7). Step 2: Connect the points in order: (4,1) to (9,1) to (9,7) to (4,7) and back to (4,1). Step 3: The shape has all right angles and opposite sides equal, so it is a rectangle. Step 4: Find the length: The x-coordinates change from 4 to 9, so length = 9 - 4 = 5 units. Step 5: Find the width: The y-coordinates change from 1 to 7, so width = 7 - 1 = 6 units. Step 6: Calculate perimeter: P = 2 × (length + width) = 2 × (5 + 6) = 2 × 11 = 22 units. The shape is a rectangle with a perimeter of 22 units.

2. 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.

3. Plot the points (7, 8), (7, 15), (18, 15), and (18, 8) on a coordinate plane. What shape is formed and what is its perimeter? Answer: rectangle, 36 Solution: Identify the shape. The points are (7,8), (7,15), (18,15), and (18,8). The x-coordinates are 7 and 18, so the horizontal distance is 18 - 7 = 11 units.
   Full step-by-step solution

   Step 1: Identify the shape. The points are (7,8), (7,15), (18,15), and (18,8). The x-coordinates are 7 and 18, so the horizontal distance is 18 - 7 = 11 units. The y-coordinates are 8 and 15, so the vertical distance is 15 - 8 = 7 units. Since opposite sides are equal and all angles are right angles, the shape is a rectangle. Step 2: Calculate the perimeter. Perimeter of a rectangle = 2 × (length + width). Here, length = 11 units and width = 7 units. So, perimeter = 2 × (11 + 7) = 2 × 18 = 36 units. The answer is rectangle, 36.

4. Liam is designing a treasure map on a coordinate plane. He marks the treasure chest at point (3, 7) and a large oak tree at point (8, 7). If he draws a straight line connecting these two points to represent a secret path, what is the length of this path? Answer: 5 Solution: We are given two points: (3, 7) and (8, 7). We need the length of the straight line between them.
   Full step-by-step solution

   We are given two points: (3, 7) and (8, 7). We need the length of the straight line between them. Step 1: Identify the coordinates Point A: (x1, y1) = (3, 7) Point B: (x2, y2) = (8, 7) Step 2: Recall the distance formula Distance = sqrt( (x2 - x1)^2 + (y2 - y1)^2 ) Step 3: Substitute the values x2 - x1 = 8 - 3 = 5 y2 - y1 = 7 - 7 = 0 Step 4: Square the differences (5)^2 = 25 (0)^2 = 0 Step 5: Add the squares 25 + 0 = 25 Step 6: Take the square root sqrt(25) = 5 Step 7: Interpret the result Since the y-coordinates are the same (both 7), the points are horizontally aligned. The horizontal distance is simply the difference in x-coordinates: 8 - 3 = 5. This matches the distance formula result. Final answer: The length of the path is 5.

5. 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.

6. Isabella is drawing a map of her neighborhood park on a coordinate plane. She marks the four corners of the rectangular playground at points A(2, 7), B(17, 7), C(17, 12), and D(2, 12). If each unit on the grid represents 1 yard, what is the perimeter of Isabella's playground in yards? Answer: 40 Solution: Find the length of the playground. Points A(2,7) and B(17,7) have the same y-coordinate, so subtract the x-coordinates: 17 - 2 = 15 yards. Find the width of the playground.
   Full step-by-step solution

   Step 1: Find the length of the playground. Points A(2,7) and B(17,7) have the same y-coordinate, so subtract the x-coordinates: 17 - 2 = 15 yards. Step 2: Find the width of the playground. Points A(2,7) and D(2,12) have the same x-coordinate, so subtract the y-coordinates: 12 - 7 = 5 yards. Step 3: A rectangle has two lengths and two widths. Perimeter = 2 × length + 2 × width = 2 × 15 + 2 × 5 = 30 + 10 = 40 yards. The perimeter of Isabella's playground is 40 yards.

7. Liam is designing a garden in the shape of a rectangle on a coordinate plane. He plots one corner at (2, 3) and the opposite corner at (8, 7). What is the area of Liam's garden in square units? Answer: 24 Solution: We are given two opposite corners of a rectangle: (2, 3) and (8, 7). These are opposite corners, so they define the length and width of the rectangle. Find the horizontal distance between the x-coordinates.
   Full step-by-step solution

   We are given two opposite corners of a rectangle: (2, 3) and (8, 7). These are opposite corners, so they define the length and width of the rectangle. Step 1: Find the horizontal distance between the x-coordinates. x1 = 2, x2 = 8 Length in x-direction = |8 - 2| = 6 Step 2: Find the vertical distance between the y-coordinates. y1 = 3, y2 = 7 Length in y-direction = |7 - 3| = 4 Step 3: Interpret these as the side lengths of the rectangle. The horizontal side length = 6 The vertical side length = 4 Step 4: Calculate the area. Area of rectangle = length × width Area = 6 × 4 = 24 Step 5: Conclusion. The area of Liam's garden is 24 square units. Answer: 24

8. A right triangle is drawn on a coordinate plane with vertices at (1, 2), (1, 7), and (5, 2). What is the area of this triangle in square units? Answer: 10 Solution: Identify the vertices: (1,2), (1,7), and (5,2). Notice that (1,2) and (1,7) have the same x-coordinate, so the distance between them is vertical: 7 - 2 = 5 units. This is one leg of the triangle.
   Full step-by-step solution

   Step 1: Identify the vertices: (1,2), (1,7), and (5,2). Step 2: Notice that (1,2) and (1,7) have the same x-coordinate, so the distance between them is vertical: 7 - 2 = 5 units. This is one leg of the triangle. Step 3: Notice that (1,2) and (5,2) have the same y-coordinate, so the distance between them is horizontal: 5 - 1 = 4 units. This is the other leg of the 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) × 4 × 5 = (1/2) × 20 = 10. The area of the triangle is 10 square units.