Distance Between Two Points
Any two points on a grid are the ends of a hidden right triangle's hypotenuse. Count across, count up, and the Pythagorean theorem does the rest. Once you can measure any segment, you can measure whole shapes.
Know these first
- Using the Pythagorean Theorem 8.G.7
- Master List skill 6.NS.8
This unlocks
- More lessons that build on this are coming.
What you'll be able to do
- Find the straight-line distance between two points, exact or rounded.
- Find the perimeter of a shape drawn on the grid.
- Use side lengths to classify a triangle or test for a right angle.
- Work backward: find a missing coordinate from a known distance.
The big idea
From one point, go straight across until you're directly above or below the other, then straight up or down to it. That path and the straight line between the points make a right triangle.
Across is the difference of the x-values; up or down is the difference of the y-values. The straight line is the hypotenuse, so distance² = across² + up².
Measuring segments unlocks shapes. Add the sides for a perimeter, compare them to classify a triangle, or test them with a² + b² = c² to find a right angle.
Words you need
- Coordinate
- A point's position, written (x, y).
- Horizontal change
- How far across: the difference of the x-values.
- Vertical change
- How far up or down: the difference of the y-values.
- Perimeter
- The total distance around a shape.
Worked examples
Example 1
Find the distance between (1, 2) and (4, 6).
- Across: 4 − 1 = 3. Up: 6 − 2 = 4.
- d² = 3² + 4² = 25, so d = 5.
Answer: 5
Example 2
Find the distance between (−3, −1) and (2, 11).
- Across: 2 − (−3) = 5. Up: 11 − (−1) = 12.
- d² = 25 + 144 = 169, so d = 13.
Answer: 13
Example 3
Find the perimeter of the triangle with vertices A(1, 1), B(4, 5) and C(7, 1).
- AB: across 3, up 4, so AB = 5.
- BC: across 3, down 4, so BC = 5.
- CA lies flat: 7 − 1 = 6.
- Perimeter = 5 + 5 + 6 = 16.
Answer: 16
Common mistakes
Subtracting with the wrong signs. Subtracting a negative means adding: 2 − (−3) = 5.
Adding the across and up distances. Walking across then up is longer than the straight line. Square, add, then take the square root.
Finding only one missing coordinate. A point at a set distance can usually be on either side - check for two answers.
Practice: 20 questions
12 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Find the distance between the two points.
- Find the distance between the two points.
- On a town map in kilometers, the library is at (2, 3) and the school is at (8, 11). How far apart are they in a straight line? Word problem
- A park map uses meters. The fountain is at (0, 0) and a bench is at (9, 12). How far is the bench from the fountain? Word problem
- A drone takes off at (−3, −2) and lands at (5, 4), in meters. How far did it fly in a straight line? Word problem
Tier 2 Intermediate One more step to think through
- Find the distance between the two points.
- Find the distance to the nearest tenth.
- Find the distance exactly, then to the nearest tenth.
- In a game, Kenji's character stands at (1, 13) and a treasure chest is at (−4, 1). How far is the chest in a straight line? Word problem
- A ship's chart in miles shows a ship at (3, 7) and a lighthouse at (11, 22). How far is the ship from the lighthouse? Word problem
- A video game character runs from (−5, −5) to (2, −1). How far is that, to the nearest tenth? Word problem
- A zip line runs from a platform at (0, 40) to a landing spot at (30, 0), in meters. How long is the line? Word problem
Tier 3 Advanced Several steps - find the hidden structure
- Find the perimeter of triangle ABC.
- Triangle ABC has vertices A(−2, 1), B(1, 5) and C(5, 2). Is it a right triangle? If so, where is the right angle?
- Three friends live at (0, 0), (6, 0) and (3, 4) on a map grid. Is the triangle between their houses equilateral, isosceles or scalene? Word problem
- A triangular hiking trail connects (1, 1), (4, 5) and (1, 5), in kilometers. How long is the whole loop? Word problem
- A garden plot has corners at (0, 0), (4, 1) and (2, 5), in meters. How much fencing goes around it, to the nearest tenth? Word problem
Tier 4 Challenge Stretch problems, some above grade level
- The point (k, 6) is 5 units from (1, 2). Find every possible value of k. Above grade level
- A dog is tied at (3, −1) on a 13-foot leash. A bone lies somewhere on the line y = 4, just at the end of the leash. Where could the bone be? Give both x-values. Word problem Above grade level
- A kite-shaped garden has corners at (0, 0), (3, 1), (4, 4) and (1, 3), in meters. How long is its border, to the nearest tenth? Word problem
Answers
Tier 1 Foundations
- 10. Across 6, up or down 8: d² = 6² + 8² = 100, so d = 10.
- 5. Across 3, up or down 4: d² = 3² + 4² = 25, so d = 5.
- 10. Across 6, up or down 8: d² = 6² + 8² = 100, so d = 10.
- 15. Across 9, up or down 12: d² = 9² + 12² = 225, so d = 15.
- 10. Across 8, up or down 6: d² = 8² + 6² = 100, so d = 10.
Tier 2 Intermediate
- 13. Across 5, up or down 12: d² = 5² + 12² = 169, so d = 13.
- √34 ≈ 5.8. Across 5, up or down 3: d² = 5² + 3² = 34, so d = √34 ≈ 5.8.
- 3√5 ≈ 6.7. Across 6, up or down 3: d² = 6² + 3² = 45, so d = 3√5 ≈ 6.7.
- 13. Across 5, up or down 12: d² = 5² + 12² = 169, so d = 13.
- 17. Across 8, up or down 15: d² = 8² + 15² = 289, so d = 17.
- √65 ≈ 8.1. Across 7, up or down 4: d² = 7² + 4² = 65, so d = √65 ≈ 8.1.
- 50. Across 30, up or down 40: d² = 30² + 40² = 2500, so d = 50.
Tier 3 Advanced
- 16. Find each side with the distance formula: 5 + 5 + 6 = 16.
- Yes - right angle at B. AB² = 25, BC² = 25, AC² = 50. The two shorter add to the longest (25 + 25 = 50), so it's a right triangle.
- Isosceles. Squared side lengths: AB² = 36, BC² = 25, AC² = 25. Exactly two match.
- 12. Find each side with the distance formula: 5 + 3 + 4 = 12.
- about 14. Find each side with the distance formula: 4.12 + 4.47 + 5.39 ≈ 14.
Tier 4 Challenge
- k = −2 or k = 4. The other leg is 4, so the missing leg is √(5² − 4²) = 3. Go that far either way from 1: k = −2 or 4.
- k = −9 or k = 15. The other leg is 5, so the missing leg is √(13² − 5²) = 12. Go that far either way from 3: k = −9 or 15.
- about 12.6. Find each side with the distance formula: 3.16 + 3.16 + 3.16 + 3.16 ≈ 12.6.
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