Coordinate Distance

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Answer Key & Explanations

Coordinate Distance · Grade 6 · Worksheet 3

1. Mere is helping to set up a rectangular vegetable garden for her community center. She marks two opposite corners of the garden on a coordinate grid: one corner at (12, 34) and the opposite corner at (12, 89). Each unit on the grid represents 2 meters. What is the length, in meters, of the side of the garden between these two points? Answer: 110 Solution: Identify the coordinates: (12, 34) and (12, 89). They have the same x-coordinate (12), so the side is vertical. Find the vertical distance in grid units: 89 - 34 = 55 units.
   Full step-by-step solution

   Step 1: Identify the coordinates: (12, 34) and (12, 89). They have the same x-coordinate (12), so the side is vertical. Step 2: Find the vertical distance in grid units: 89 - 34 = 55 units. Step 3: Each unit represents 2 meters, so multiply: 55 * 2 = 110 meters. The length of the garden side is 110 meters.

2. Matiu is helping his school's art club create a giant mural on a coordinate grid painted on the playground. Each unit on the grid represents 1 meter. He places two large star stickers at points (15, 42) and (15, 89) to mark where the top and bottom of a rainbow should go. What is the vertical distance, in meters, between the two star stickers? Answer: 47 Solution: Identify the coordinates. The first star is at (15, 42) and the second star is at (15, 89). They share the same x-coordinate (15), so the distance is vertical.
   Full step-by-step solution

   Step 1: Identify the coordinates. The first star is at (15, 42) and the second star is at (15, 89). They share the same x-coordinate (15), so the distance is vertical. Step 2: Find the difference in the y-coordinates: 89 - 42 = 47. Step 3: Since each unit represents 1 meter, the vertical distance is 47 meters. The answer is 47.

3. (-3, 4) and (7, -2) = ? Answer: √136 Solution: Identify the coordinates: Point 1 is (-3, 4) and Point 2 is (7, -2). Substitute the values: distance = sqrt((7 - (-3))^2 + (-2 - 4)^2). Simplify inside the parentheses: distance = sqrt((10)^2 + (-6)^2).
   Full step-by-step solution

   Step 1: Identify the coordinates: Point 1 is (-3, 4) and Point 2 is (7, -2). Step 2: Apply the distance formula: distance = sqrt((x2 - x1)^2 + (y2 - y1)^2). Step 3: Substitute the values: distance = sqrt((7 - (-3))^2 + (-2 - 4)^2). Step 4: Simplify inside the parentheses: distance = sqrt((10)^2 + (-6)^2). Step 5: Square the numbers: distance = sqrt(100 + 36). Step 6: Add the results: distance = sqrt(136). The final answer is √136.

4. √(256) + (-5)³ × 2 ÷ 10 = ? Answer: -9 Solution: Calculate the square root: √(256) = 16 Calculate the exponent: (-5)³ = -5 × -5 × -5 = 25 × -5 = -125 Perform multiplication and division from left to right: -125 × 2 = -250, then -250 ÷ 10 = -25 Add the results: 16 + (-25) = 16 - 25 = -9 The answer is -9.
   Full step-by-step solution

   Step 1: Calculate the square root: √(256) = 16 Step 2: Calculate the exponent: (-5)³ = -5 × -5 × -5 = 25 × -5 = -125 Step 3: Perform multiplication and division from left to right: -125 × 2 = -250, then -250 ÷ 10 = -25 Step 4: Add the results: 16 + (-25) = 16 - 25 = -9 The answer is -9.

5. Distance from (-15, 20) to (35, 20)? Answer: 50 Solution: Identify the coordinates. Point A is (-15, 20) and Point B is (35, 20). They share the same y-coordinate (20), so the distance is horizontal.
   Full step-by-step solution

   Step 1: Identify the coordinates. Point A is (-15, 20) and Point B is (35, 20). They share the same y-coordinate (20), so the distance is horizontal. Step 2: Find the difference in x-coordinates: 35 - (-15) = 35 + 15 = 50. Step 3: The distance is the absolute value of the difference: |50| = 50. The answer is 50.

6. Distance from (17, -27) to (17, 12)? Answer: 39 Solution: Identify the coordinates. Point A is (17, -27) and Point B is (17, 12). They have the same x-coordinate (17), so the distance is vertical.
   Full step-by-step solution

   Step 1: Identify the coordinates. Point A is (17, -27) and Point B is (17, 12). They have the same x-coordinate (17), so the distance is vertical. Step 2: Find the difference in y-coordinates: 12 - (-27) = 12 + 27 = 39. Step 3: Take the absolute value: |39| = 39. The distance between (17, -27) and (17, 12) is 39 units.

7. Liam is designing a rectangular garden for his school. He places the four corners at coordinates (2, 5), (2, 12), (9, 12), and (9, 5) on a coordinate grid where each unit represents 1 meter. To protect the garden from rabbits, he needs to install a fence along the entire perimeter. What is the total length of fencing, in meters, that Liam needs? Answer: 28 Solution: A = (2, 5) B = (2, 12) C = (9, 12) D = (9, 5) A and B have the same x-coordinate (2), so AB is vertical. B and C have the same y-coordinate (12), so BC is horizontal.
   Full step-by-step solution

   Let's solve step by step. --- **Step 1: Identify the coordinates and shape** The four corners are: A = (2, 5) B = (2, 12) C = (9, 12) D = (9, 5) Plotting these points: A and B have the same x-coordinate (2), so AB is vertical. B and C have the same y-coordinate (12), so BC is horizontal. C and D have the same x-coordinate (9), so CD is vertical. D and A have the same y-coordinate (5), so DA is horizontal. This is a rectangle. --- **Step 2: Find the length of the sides** Vertical side AB: A = (2, 5), B = (2, 12) Length = |12 - 5| = 7 meters. Horizontal side BC: B = (2, 12), C = (9, 12) Length = |9 - 2| = 7 meters. Vertical side CD: C = (9, 12), D = (9, 5) Length = |12 - 5| = 7 meters. Horizontal side DA: D = (9, 5), A = (2, 5) Length = |9 - 2| = 7 meters. --- **Step 3: Check rectangle dimensions** Actually, AB = 7 m (vertical), BC = 7 m (horizontal), CD = 7 m (vertical), DA = 7 m (horizontal). So all sides are 7 m — this is a square. --- **Step 4: Calculate perimeter** Perimeter of a rectangle = 2 × (length + width) Here length = 7, width = 7. Perimeter = 2 × (7 + 7) = 2 × 14 = 28 meters. --- **Step 5: Conclusion** Liam needs 28 meters of fencing. --- **Final answer:** 28

8. Distance from (15, -23) to (15, 34)? Answer: 57 Solution: Identify the coordinates. Point A is (15, -23) and Point B is (15, 34). The x-coordinates are both 15, so the points are vertically aligned.
   Full step-by-step solution

   Step 1: Identify the coordinates. Point A is (15, -23) and Point B is (15, 34). The x-coordinates are both 15, so the points are vertically aligned. Step 2: Find the difference in y-coordinates: 34 - (-23) = 34 + 23 = 57. Step 3: Take the absolute value: |57| = 57. The distance between the points is 57 units.