The Pythagorean Theorem & Distance on the Coordinate Plane
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This is a paper math packet about the Pythagorean Theorem and using it to find distance on the coordinate plane. In a right triangle, the two shorter sides (the legs) and the longest side (the hypotenuse) are connected by the rule a² + b² = c². You will use it to find a missing side, to measure the distance between two plotted points, to solve a real-world problem (a ladder), to fix an error, and to test whether a triangle is a right triangle (the converse). Work by yourself with a pencil. A calculator is not required, but a basic or scientific calculator is allowed if you have one. Show your work — every answer should show the squares you computed and the square root you took, not just a final number. Give both an exact answer (such as √52) and a rounded answer (to the nearest tenth) when a number is not a perfect square. Everything you need (formulas, a worked example, a labeled graph) is printed right here. If a question is hard, skip it, keep going, and come back. Circle or box your final answers.
Start Warm-Up: Retrieval 6 min
Bring back squaring, square roots, and reading points off a grid.
1Square and square-root. Find 5² and 12², then add them. Finally, find √169. Show each step. (Remember: 5² means 5 × 5, not 5 × 2.)
2Read a horizontal and a vertical distance. A point starts at (2, 3) and a second point is at (2, 8). How far apart are they up-and-down (the vertical distance)? Then a third point is at (6, 3): how far is it from (2, 3) left-and-right (the horizontal distance)? Show the subtraction you used.
Build Study the Math 8–12 min
What you need to know (with formulas)
A right triangle has one 90° angle (a square corner). The two sides that form the right angle are the legs (call their lengths a and b). The side across from the right angle is the hypotenuse (length c). The hypotenuse is always the longest side — that is how you can spot it.
The Pythagorean Theorem. For any right triangle:
a² + b² = c²
The two legs squared and added equal the hypotenuse squared. Use it two ways:
- Missing hypotenuse (c): add the squares of the legs, then take the square root: c = √(a² + b²).
- Missing leg (a or b): the hypotenuse is already known, so subtract: a = √(c² − b²). (Square the hypotenuse, subtract the known leg's square, then take the square root.)
The Distance Formula (an application). To find the distance between two points (x₁, y₁) and (x₂, y₂) on a grid, make a right triangle: the horizontal leg is the change in x and the vertical leg is the change in y. Then apply the theorem:
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
This is just a² + b² = c² where a = (x₂ − x₁), b = (y₂ − y₁), and d = c. Subtracting in the other order is fine because squaring makes it positive either way.
The Converse (a right-triangle test). The theorem also works backward. If the three side lengths of a triangle make a² + b² = c² true (with c the longest side), then the triangle is a right triangle. If a² + b² ≠ c², it is not a right triangle. To use it: square all three sides, add the squares of the two shorter sides, and compare that sum to the square of the longest side.
Tip: when a² + b² is not a perfect square, the exact answer is a square root (like √52). Leave it as a root for the exact value, and also give a rounded decimal (√52 ≈ 7.2) to the nearest tenth.
Worked Example — a 6-8-10 right triangle on the grid
Look at Figure 1. The right angle is at B(7, 1). One leg runs from A(1, 1) to B(7, 1); the other leg runs from B(7, 1) to C(7, 9). The hypotenuse is the slanted side from A(1, 1) to C(7, 9) — it is across from the right angle and is the longest side.
Find the legs by counting boxes: leg a = AB = 7 − 1 = 6 units (horizontal); leg b = BC = 9 − 1 = 8 units (vertical).
Find the hypotenuse c with the theorem:
a² + b² = c² → 6² + 8² = c² → 36 + 64 = c² → 100 = c²
c = √100 = 10. So the hypotenuse AC is 10 units. (Exact and rounded agree here because 100 is a perfect square.)
Same answer using the distance formula for A(1, 1) and C(7, 9): d = √[(7 − 1)² + (9 − 1)²] = √[6² + 8²] = √[36 + 64] = √100 = 10. ✔
Check the converse: the sides are 6, 8, 10, and 10 is longest. 6² + 8² = 36 + 64 = 100, and 10² = 100. Since 100 = 100, the triangle is a right triangle. ✔
| Side | From → To | Type | Length (units) |
|---|---|---|---|
| AB | (1, 1) → (7, 1) | leg a (horizontal) | 6 |
| BC | (7, 1) → (7, 9) | leg b (vertical) | 8 |
| AC | (1, 1) → (7, 9) | hypotenuse c | 10 |
3Using Figure 1, suppose the vertical leg BC were 12 instead of 8 (with the horizontal leg AB still 6). Use a² + b² = c² to find the new hypotenuse. Give the exact answer (a square root if needed) and the answer rounded to the nearest tenth. Show your work.
Apply Use What You Know 20–26 min
Show the squares and the square root in every item. A calculator is allowed. When the answer is not a perfect square, give both the exact root and the rounded decimal.
4Find a missing side.
4a. Missing hypotenuse. A right triangle has legs a = 9 and b = 12. Find the hypotenuse c. Use a² + b² = c² and show every step. (This one comes out to a whole number.)
4b. Missing leg. A different right triangle has one leg b = 5 and hypotenuse c = 13. Find the other leg a. (Careful: the hypotenuse is known, so you must subtract: a² = c² − b².) Show your steps.
5Distance between two plotted points. On a coordinate grid, point P is at (1, 2) and point Q is at (7, 10). Find the distance PQ. Use the distance formula d = √[(x₂ − x₁)² + (y₂ − y₁)²]; show the horizontal change, the vertical change, both squares, the sum, and the square root.
6Real-world application (a ladder). A 17-foot ladder leans against a wall. Its base is 8 feet from the bottom of the wall. How high up the wall does the ladder reach? The wall and ground make the right angle, the ground distance and the wall height are the legs, and the ladder is the hypotenuse. Draw a quick right triangle, label it, and solve. Show your work.
7Error analysis (added instead of using the theorem). A student found the hypotenuse of a right triangle with legs a = 6 and b = 8 and wrote the work shown. Find the mistake, explain it, and give the correct hypotenuse.
Student's work (legs 6 and 8):
c = a + b = 6 + 8 = 14. So the hypotenuse is 14.
What is wrong, and what is the correct hypotenuse? Fix it in the box, showing the correct use of a² + b² = c².
8Use the converse to test a triangle. A triangle has side lengths 10, 24, and 26. Is it a right triangle? Identify the longest side, square all three sides, and check whether a² + b² = c². State your conclusion clearly (yes or no) and why.
Block Extra Applied Task block only · ~12–15 min
9Multi-step composite distance. A hiker walks a path with three stops plotted on a grid (each unit = 1 kilometer): R(0, 0), then S(3, 4), then T(10, 4). She walks a straight segment from R to S, then a straight segment from S to T.
9a. Find the length of segment RS using the distance formula. Show your work.
9b. Find the length of segment ST using the distance formula. Show your work.
9c. Find the total distance walked (RS + ST). Then find the straight-line distance from R directly to T (the distance formula for R to T). Which is longer, the path or the straight line? Explain in one sentence why that makes sense.
Explain Justify Your Strategy 6–10 min
Question 10. A triangle has side lengths 7, 9, and 12. A classmate says, "It must be a right triangle because it has three different sides." Is the triangle a right triangle? Decide, and justify your answer using the converse of the Pythagorean Theorem. Write a Claim (your yes/no answer), Evidence (the squaring and adding you computed), and Reasoning (why that computation proves your claim, and why the classmate's reason is not enough).
Sentence stems you may use: "The triangle is / is not a right triangle because…" · "The longest side is ___, so c = ___." · "I found a² + b² = ___ and c² = ___." · "Since a² + b² ___ c², the converse tells me…" · "Having three different sides does not prove a right angle because…"
Close ACE Wrap-Up 5 min
In your own words, explain how you can tell which side of a right triangle is the hypotenuse, and why you must square-root at the end instead of stopping at c².
Point to one item in this packet where the distance formula was really just a² + b² = c² in disguise. Give its number and name the legs.
Give a new real-life situation (not the ladder) where you could use the Pythagorean Theorem to find a distance you cannot measure directly.
Continue Early Finisher optional
Find a Pythagorean triple of your own. A Pythagorean triple is three whole numbers a, b, c with a² + b² = c² (like 3-4-5 or 6-8-10). On the back of this page: (1) start with the triple 3-4-5 and multiply all three numbers by the same whole number (try 2, then 3) to make two new triples; (2) prove each new set really works by showing a² + b² = c²; and (3) explain in one sentence why multiplying a triple by the same number always gives another right triangle. Then use the converse to show that 5-6-8 is not a right triangle.