Coordinate Distance

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Answer Key & Explanations

Coordinate Distance · Grade 6 · Worksheet 1

1. Distance from (16, -21) to (16, 36)? Answer: 57 Solution: Identify the coordinates. The points are (16, -21) and (16, 36). They have the same x-coordinate (16), so the distance is the absolute difference of the y-coordinates.
   Full step-by-step solution

   Step 1: Identify the coordinates. The points are (16, -21) and (16, 36). They have the same x-coordinate (16), so the distance is the absolute difference of the y-coordinates. Step 2: Subtract the y-coordinates: 36 - (-21) = 36 + 21 = 57. Step 3: Take the absolute value: |57| = 57. The answer is 57.

2. Noah is designing a new running track for his school's field day. He marks the starting point at (8, 15) and the finish line at (24, 39) on a coordinate grid where each unit represents 5 meters. What is the actual distance, in meters, between the starting point and finish line of Noah's running track? Answer: 130 Solution: Identify the coordinates: Starting point (8, 15) and Finish line (24, 39) Calculate the horizontal distance: 24 - 8 = 16 units Calculate the vertical distance: 39 - 15 = 24 units Use the Pythagorean theorem to find the straight-line distance: distance = sqrt(16^2 + 24^2) Calculate 16^2 = 256 and…
   Full step-by-step solution

   Step 1: Identify the coordinates: Starting point (8, 15) and Finish line (24, 39) Step 2: Calculate the horizontal distance: 24 - 8 = 16 units Step 3: Calculate the vertical distance: 39 - 15 = 24 units Step 4: Use the Pythagorean theorem to find the straight-line distance: distance = sqrt(16^2 + 24^2) Step 5: Calculate 16^2 = 256 and 24^2 = 576 Step 6: Add the squares: 256 + 576 = 832 Step 7: Find the square root: sqrt(832) = sqrt(16 * 52) = 4 * sqrt(52) = 4 * sqrt(4 * 13) = 8 * sqrt(13) units Step 8: Since each unit represents 5 meters, multiply by 5: 8 * sqrt(13) * 5 = 40 * sqrt(13) meters Step 9: Calculate the numerical value: sqrt(13) ≈ 3.60555, so 40 * 3.60555 ≈ 130.222 Step 10: Round to the nearest meter: 130 meters Noah's running track is approximately 130 meters long.

3. (-8)² - 12 ÷ (-3) = ? Answer: 68 Solution: Evaluate the exponent first: (-8)² = (-8) × (-8) = 64 Evaluate the division: 12 ÷ (-3) = -4 Substitute the results back into the expression: 64 - (-4) Subtracting a negative is the same as adding a positive: 64 + 4 = 68 The answer is 68.
   Full step-by-step solution

   Step 1: Evaluate the exponent first: (-8)² = (-8) × (-8) = 64 Step 2: Evaluate the division: 12 ÷ (-3) = -4 Step 3: Substitute the results back into the expression: 64 - (-4) Step 4: Subtracting a negative is the same as adding a positive: 64 + 4 = 68 The answer is 68.

4. Distance from (12, -24) to (12, 18) = ? Answer: 42 Solution: Identify the coordinates. Point A is (12, -24) and Point B is (12, 18). The x-coordinates are the same (12), so the distance is vertical.
   Full step-by-step solution

   Step 1: Identify the coordinates. Point A is (12, -24) and Point B is (12, 18). The x-coordinates are the same (12), so the distance is vertical. Step 2: Find the difference in y-coordinates: 18 - (-24) = 18 + 24 = 42. Step 3: Take the absolute value: |42| = 42. The distance between the points is 42 units.

5. Noah is helping his school's robotics team plan a straight-line path for their robot on a coordinate grid. The robot starts at point (6, 11) and moves directly east to point (6, 41). Each unit on the grid represents 1 meter. What is the total distance, in meters, the robot travels? Answer: 30 Solution: Identify the coordinates: start (6, 11) and end (6, 41). Notice the x-coordinates are the same (6), so the robot moves vertically. Find the difference in y-coordinates: 41 - 11 = 30.
   Full step-by-step solution

   Step 1: Identify the coordinates: start (6, 11) and end (6, 41). Step 2: Notice the x-coordinates are the same (6), so the robot moves vertically. Step 3: Find the difference in y-coordinates: 41 - 11 = 30. Step 4: Since each unit is 1 meter, the distance is 30 meters. The answer is 30.

6. A rectangular garden is plotted on a coordinate plane with corners at (2, 1), (8, 1), (8, 5), and (2, 5). A straight stone path runs diagonally from the bottom-left corner to the top-right corner. What is the length of this path in meters? Answer: 7.2 Solution: Identify the coordinates of the bottom-left and top-right corners. The garden's corners are (2, 1), (8, 1), (8, 5), (2, 5). Bottom-left corner: (2, 1) Top-right corner: (8, 5) Understand the path.
   Full step-by-step solution

   Let's solve this step-by-step. Step 1: Identify the coordinates of the bottom-left and top-right corners. The garden's corners are (2, 1), (8, 1), (8, 5), (2, 5). Bottom-left corner: (2, 1) Top-right corner: (8, 5) Step 2: Understand the path. The path runs diagonally from (2, 1) to (8, 5). Step 3: Use the distance formula. Distance between two points (x1, y1) and (x2, y2) is: d = sqrt( (x2 - x1)^2 + (y2 - y1)^2 ) Step 4: Substitute the coordinates. x1 = 2, y1 = 1 x2 = 8, y2 = 5 d = sqrt( (8 - 2)^2 + (5 - 1)^2 ) d = sqrt( (6)^2 + (4)^2 ) d = sqrt( 36 + 16 ) d = sqrt( 52 ) Step 5: Simplify sqrt(52). 52 = 4 * 13 sqrt(52) = sqrt(4 * 13) = sqrt(4) * sqrt(13) = 2 * sqrt(13) Step 6: Approximate the numerical value. sqrt(13) ≈ 3.60555 So d ≈ 2 * 3.60555 = 7.2111 Step 7: Round to one decimal place as in the given correct answer. 7.2111 → 7.2 Final Answer: The length of the path is 7.2 meters.

7. Mere is helping her school's art club create a large mural on a coordinate grid where each unit represents 2 meters. She marks the location for a tree at point (120, 84) and a flower bed at point (120, 48). What is the actual distance, in meters, between the tree and the flower bed? Answer: 72 Solution: The two points share the same x-coordinate (120), so the distance is the absolute difference in y-coordinates. Vertical difference = |84 - 48| = 36 units.
   Full step-by-step solution

   Step 1: The two points share the same x-coordinate (120), so the distance is the absolute difference in y-coordinates. Step 2: Vertical difference = |84 - 48| = 36 units. Step 3: Each unit represents 2 meters, so the actual distance = 36 × 2 = 72 meters. The answer is 72 meters.

8. Distance from (-13, 25) to (21, 25) = ? Answer: 34 Solution: Identify the coordinates. Point A is (-13, 25) and Point B is (21, 25). Since the y-coordinates are both 25, the distance is the absolute difference of the x-coordinates.
   Full step-by-step solution

   Step 1: Identify the coordinates. Point A is (-13, 25) and Point B is (21, 25). Since the y-coordinates are both 25, the distance is the absolute difference of the x-coordinates. Step 2: Subtract the x-coordinates: 21 - (-13) = 21 + 13 = 34. Step 3: Take the absolute value: |34| = 34. The distance between the points is 34 units.

9. Distance from (15, -25) to (15, 35) = ? Answer: 60 Solution: Identify the coordinates. The points are (15, -25) and (15, 35). Since the x-coordinates are both 15, the distance is vertical.
   Full step-by-step solution

   Step 1: Identify the coordinates. The points are (15, -25) and (15, 35). Since the x-coordinates are both 15, the distance is vertical. Step 2: Find the difference in y-coordinates: 35 - (-25) = 35 + 25 = 60. Step 3: Take the absolute value: |60| = 60. The answer is 60.