Coordinate Distance

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Coordinate Distance · Grade 6 · Worksheet 3

1. Distance from (15, -27) to (15, 42)? Answer: 69 Solution: Identify the coordinates. Point A is (15, -27) and Point B is (15, 42). They have the same x-coordinate (15), so the distance is vertical.
   Full step-by-step solution

   Step 1: Identify the coordinates. Point A is (15, -27) and Point B is (15, 42). They have the same x-coordinate (15), so the distance is vertical. Step 2: Find the difference in y-coordinates: 42 - (-27) = 42 + 27 = 69. Step 3: The distance is the absolute value of the difference: |69| = 69. The answer is 69.

2. A rectangular sports field is mapped on a coordinate plane with corners at (120, 50), (480, 50), (480, 290), and (120, 290). The grounds crew needs to install a straight underground irrigation pipe that runs diagonally from the southwest corner to the northeast corner. What is the length of this diagonal irrigation pipe in meters? Answer: 410 Solution: Identify the coordinates of the southwest and northeast corners. Southwest corner is (120, 50) and northeast corner is (480, 290). Calculate the horizontal distance (length) between these points: 480 - 120 = 360 meters.
   Full step-by-step solution

   Step 1: Identify the coordinates of the southwest and northeast corners. Southwest corner is (120, 50) and northeast corner is (480, 290). Step 2: Calculate the horizontal distance (length) between these points: 480 - 120 = 360 meters. Step 3: Calculate the vertical distance (width) between these points: 290 - 50 = 240 meters. Step 4: Use the Pythagorean theorem to find the diagonal length: diagonal = sqrt(360^2 + 240^2). Step 5: Calculate 360^2 = 129,600 and 240^2 = 57,600. Step 6: Add these squares: 129,600 + 57,600 = 187,200. Step 7: Take the square root: sqrt(187,200) = 410. The length of the diagonal irrigation pipe is 410 meters.

3. Liam is planning a bike route on a coordinate grid. He starts at his house at point (3, 7) and rides to the library at point (9, 15). If each unit on the grid represents 250 meters, what is the total distance, in meters, that Liam rides? Answer: 2500 Solution: Find the horizontal distance between the two points. Liam starts at (3, 7) and ends at (9, 15). Horizontal difference = 9 - 3 = 6 units.
   Full step-by-step solution

   Step 1: Find the horizontal distance between the two points. Liam starts at (3, 7) and ends at (9, 15). Horizontal difference = 9 - 3 = 6 units. Step 2: Find the vertical distance between the two points. Vertical difference = 15 - 7 = 8 units. Step 3: Use the Pythagorean theorem to find the straight-line distance in grid units. Distance in units = square root of (horizontal difference squared + vertical difference squared) = square root of (6^2 + 8^2) = square root of (36 + 64) = square root of (100) = 10 units. Step 4: Convert grid units to meters. Each unit represents 250 meters. Total distance = 10 units × 250 meters/unit = 2500 meters. Final answer: 2500 meters.

4. Matiu is helping to set up a treasure hunt for his school's cultural festival. He places the first clue at point (6, 1024) on a coordinate grid where each unit represents 1 meter. The second clue is directly south at point (6, 978). What is the distance, in meters, between the two clues? Answer: 46 Solution: Identify the coordinates. First clue: (6, 1024). Second clue: (6, 978).
   Full step-by-step solution

   Step 1: Identify the coordinates. First clue: (6, 1024). Second clue: (6, 978). Step 2: Since the x-coordinates are the same (both 6), the points lie on a vertical line. The distance is the absolute difference in the y-coordinates. Step 3: Calculate the difference: 1024 - 978 = 46. Step 4: The absolute value of 46 is 46. The distance between the two clues is 46 meters.

5. A rectangular city park is mapped on a coordinate plane with corners at (150, 80), (450, 80), (450, 320), and (150, 320). The city plans to build a straight bike path directly from the southwest corner to the northeast corner. What is the length of this diagonal bike path in meters? Answer: 370 Solution: Identify the coordinates of the southwest and northeast corners. Southwest corner is (150, 80) and northeast corner is (450, 320). Calculate the horizontal distance (width) of the rectangle: 450 - 150 = 300 meters.
   Full step-by-step solution

   Step 1: Identify the coordinates of the southwest and northeast corners. Southwest corner is (150, 80) and northeast corner is (450, 320). Step 2: Calculate the horizontal distance (width) of the rectangle: 450 - 150 = 300 meters. Step 3: Calculate the vertical distance (height) of the rectangle: 320 - 80 = 240 meters. Step 4: Use the Pythagorean theorem to find the diagonal length: diagonal^2 = width^2 + height^2 Step 5: diagonal^2 = 300^2 + 240^2 = 90,000 + 57,600 = 147,600 Step 6: diagonal = sqrt(147,600) = 370 Step 7: The length of the diagonal bike path is 370 meters.

6. Liam is tracking the temperature changes in his city over a week. On Monday morning, the temperature was 12°C. By afternoon, it rose by 8°C. Overnight, it dropped by 15°C. What was the temperature on Tuesday morning? Answer: 5°C Solution: Start with Monday morning temperature. Monday morning temperature = 12°C. Temperature rise by afternoon.
   Full step-by-step solution

   Let's go step by step. Step 1: Start with Monday morning temperature. Monday morning temperature = 12°C. Step 2: Temperature rise by afternoon. It rose by 8°C, so: Monday afternoon temperature = 12 + 8 = 20°C. Step 3: Overnight drop. From Monday afternoon to Tuesday morning, it dropped by 15°C. So: Tuesday morning temperature = 20 - 15 = 5°C. Step 4: Conclusion. The temperature on Tuesday morning was 5°C.

7. Distance from (Aroha, 15) to (Aroha, 27)? Answer: 12 Solution: Identify the coordinates. Both points have the same x-coordinate (Aroha), so they lie on a vertical line. Find the y-coordinates: 15 and 27.
   Full step-by-step solution

   Step 1: Identify the coordinates. Both points have the same x-coordinate (Aroha), so they lie on a vertical line. Step 2: Find the y-coordinates: 15 and 27. Step 3: Calculate the difference: 27 - 15 = 12. Step 4: Distance is always positive, so the distance is 12. The answer is 12.

8. Matiu is helping his school's art club create a giant mural on a coordinate grid. They paint a star at point (42, 58) and a moon at point (42, 16) on the grid. Each unit on the grid represents 2 meters. What is the actual distance, in meters, between the star and the moon? Answer: 84 Solution: Identify the coordinates. Star: (42, 58). Moon: (42, 16).
   Full step-by-step solution

   Step 1: Identify the coordinates. Star: (42, 58). Moon: (42, 16). They have the same x-coordinate (42), so they are vertically aligned. Step 2: Find the vertical distance in grid units by subtracting the y-coordinates: 58 - 16 = 42 units. Step 3: Each unit represents 2 meters. Multiply the number of units by 2: 42 units * 2 meters/unit = 84 meters. The actual distance between the star and the moon is 84 meters.