Coordinate Distance

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Answer Key & Explanations

Coordinate Distance · Grade 6 · Worksheet 2

1. √(144) + |−15| − 3² = ? Answer: 18 Solution: √(144) + |−15| − 3² √(144) = 12 Because 12 × 12 = 144. |−15| = 15 Because absolute value makes any negative number positive. 3² = 9 Because 3 × 3 = 9.
   Full step-by-step solution

   Let's solve step by step. We have: √(144) + |−15| − 3² **Step 1: Evaluate the square root** √(144) = 12 Because 12 × 12 = 144. **Step 2: Evaluate the absolute value** |−15| = 15 Because absolute value makes any negative number positive. **Step 3: Evaluate the exponent** 3² = 9 Because 3 × 3 = 9. **Step 4: Substitute back into the expression** 12 + 15 − 9 **Step 5: Perform addition first** 12 + 15 = 27 **Step 6: Perform subtraction** 27 − 9 = 18 **Final Answer:** 18

2. Olivia is helping her school's art club paint a large mural on a coordinate grid wall. She paints a blue star at point (101, 73) and a red star directly below it at point (101, 53). Each unit on the grid represents 1 centimeter. What is the distance, in centimeters, between the blue star and the red star? Answer: 20 Solution: Identify the coordinates: Blue star (101, 73) and Red star (101, 53). Notice both points have the same x-coordinate (101), so they are aligned vertically. Find the difference in the y-coordinates: 73 - 53 = 20.
   Full step-by-step solution

   Step 1: Identify the coordinates: Blue star (101, 73) and Red star (101, 53). Step 2: Notice both points have the same x-coordinate (101), so they are aligned vertically. Step 3: Find the difference in the y-coordinates: 73 - 53 = 20. Step 4: Since each unit equals 1 centimeter, the distance is 20 centimeters. Final answer: 20 centimeters.

3. 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.

4. A rectangular city park is mapped on a coordinate plane with corners at (1, 2), (9, 2), (9, 8), and (1, 8). The park department wants to install a straight bike path that runs diagonally from the southwest corner to the northeast corner. What is the length of this bike path in meters? Round your answer to the nearest tenth of a meter. Answer: 10.0 Solution: Identify the coordinates of the southwest and northeast corners. Southwest corner: (1, 2), Northeast corner: (9, 8). Calculate the horizontal distance (length) of the rectangle.
   Full step-by-step solution

   Step 1: Identify the coordinates of the southwest and northeast corners. Southwest corner: (1, 2), Northeast corner: (9, 8). Step 2: Calculate the horizontal distance (length) of the rectangle. 9 - 1 = 8 meters. Step 3: Calculate the vertical distance (width) of the rectangle. 8 - 2 = 6 meters. Step 4: The diagonal path forms the hypotenuse of a right triangle with legs of 8 m and 6 m. Step 5: Apply the Pythagorean theorem: diagonal^2 = 8^2 + 6^2. Step 6: Calculate: 8^2 = 64, 6^2 = 36. Step 7: Add: 64 + 36 = 100. Step 8: Find the square root: sqrt(100) = 10. Step 9: The problem asks to round to the nearest tenth. 10.0 is already at the nearest tenth. The length of the bike path is 10.0 meters.

5. √(169) + |−15| = ? Answer: 28 Solution: We need to evaluate √(169) + |−15|. Evaluate the square root. √(169) means the positive square root of 169.
   Full step-by-step solution

   Step 1: Understand the expression. We need to evaluate √(169) + |−15|. Step 2: Evaluate the square root. √(169) means the positive square root of 169. Since 13 × 13 = 169, we have √(169) = 13. Step 3: Evaluate the absolute value. |−15| means the distance from −15 to 0 on the number line, which is 15. So |−15| = 15. Step 4: Add the results. 13 + 15 = 28. Step 5: Final answer. The result is 28.

6. Liam is helping his school's art club paint a mural on a large coordinate grid on the playground wall. The grid uses units of feet. The club's design includes a vertical stripe that goes from point (5, 11) to point (5, 43). What is the length of this vertical stripe in feet? Answer: 32 Solution: Identify the coordinates: Point 1 = (5, 11), Point 2 = (5, 43). Notice both points have the same x-coordinate (5), so the distance is vertical. Find the difference in y-coordinates: 43 - 11 = 32.
   Full step-by-step solution

   Step 1: Identify the coordinates: Point 1 = (5, 11), Point 2 = (5, 43). Step 2: Notice both points have the same x-coordinate (5), so the distance is vertical. Step 3: Find the difference in y-coordinates: 43 - 11 = 32. Step 4: The distance between the points is 32 units. Step 5: Since each unit represents 1 foot, the stripe is 32 feet long. Answer: 32

7. Olivia is helping her town create a new nature trail on a coordinate map. The trail starts at the entrance gate at (15, 23) and goes straight to a bird-watching platform at (15, 67). Each unit on the map represents 3 meters. What is the actual distance, in meters, between the entrance gate and the bird-watching platform? Answer: 132 Solution: Identify the coordinates. Entrance gate: (15, 23), Bird-watching platform: (15, 67). Both points have the same x-coordinate (15), so they are aligned vertically.
   Full step-by-step solution

   Step 1: Identify the coordinates. Entrance gate: (15, 23), Bird-watching platform: (15, 67). Both points have the same x-coordinate (15), so they are aligned vertically. Step 2: Find the vertical distance by subtracting the y-coordinates: 67 - 23 = 44 units. Step 3: Each unit on the map represents 3 meters. Multiply the distance in units by the scale factor: 44 units x 3 meters per unit = 132 meters. The answer is 132 meters.

8. Distance from (12, -18) to (12, 35) = ? Answer: 53 Solution: Identify the coordinates. Point A is (12, -18) and Point B is (12, 35). They have the same x-coordinate (12), so the distance is vertical.
   Full step-by-step solution

   Step 1: Identify the coordinates. Point A is (12, -18) and Point B is (12, 35). They have the same x-coordinate (12), so the distance is vertical. Step 2: Find the difference in y-coordinates: 35 - (-18) = 35 + 18 = 53. Step 3: Take the absolute value: |53| = 53. The distance between the points is 53 units.