The Pythagorean Theorem & Distance on the Coordinate Plane

Overview

This is a self-contained, low-tech substitute packet in which students practice the Pythagorean Theorem and use it to find distance on the coordinate plane. Working alone with a pencil (a basic or scientific calculator is allowed but not required), students warm up with squaring and square roots, study a worked example built on a labeled 6-8-10 right triangle, and then apply the ideas: finding a missing leg or hypotenuse, finding the distance between two plotted points, solving a real-world ladder problem, fixing an error where a student added the legs, and using the converse to test whether a triangle is a right triangle. On a block schedule they also work a multi-step composite-distance task. They finish by justifying, with a claim, evidence, and reasoning, whether a given triangle is right. Throughout, students give both an exact answer (a square root when needed) and a rounded decimal.

Accessible version of the student activity

The full student activity is reproduced below in plain, screen-reader-friendly HTML. It reflows on phones and at 200% zoom. Write your answers on the printed packet, and show your work.

Formulas you will use

• Pythagorean Theorem (right triangles): a² + b² = c², where a and b are the legs and c is the hypotenuse (the side across from the right angle, always the longest side).
• Missing hypotenuse: c = √(a² + b²). Missing leg: a = √(c² − b²).
• Distance formula (two points (x₁, y₁) and (x₂, y₂)): d = √[(x₂ − x₁)² + (y₂ − y₁)²]. This is a² + b² = c² with the horizontal change and vertical change as the legs.
• Converse: if a² + b² = c² for a triangle's sides (c the longest), the triangle is a right triangle; if a² + b² ≠ c², it is not.

Start (6 minutes) — Retrieval warm-up

1. Find 5² and 12², add them, then find √169. Show each step. (5² means 5 × 5, not 5 × 2.)
2. Find the vertical distance from (2, 3) to (2, 8), and the horizontal distance from (2, 3) to (6, 3). Show the subtraction.

Build (8–12 minutes) — Study the math

A right triangle has a 90° angle. The two sides forming it are the legs (a and b); the side across from it is the hypotenuse (c), always the longest side. The Pythagorean Theorem says a² + b² = c². To find the hypotenuse, add the squares of the legs and take the square root: c = √(a² + b²). To find a missing leg when the hypotenuse is known, subtract: a = √(c² − b²). The distance formula, d = √[(x₂ − x₁)² + (y₂ − y₁)²], is the same theorem, using the horizontal change and vertical change between two points as the two legs. The converse tests a triangle: square all three sides; if the two shorter squares add to the longest square (a² + b² = c²), the triangle is right, otherwise it is not. When a² + b² is not a perfect square, give the exact answer as a square root (like √52) and also a rounded decimal to the nearest tenth.

Worked example — a 6-8-10 right triangle on the grid

A right triangle has vertices A(1, 1), B(7, 1), and C(7, 9), with the right angle at B. The horizontal leg AB has length 7 − 1 = 6; the vertical leg BC has length 9 − 1 = 8; the hypotenuse is the slanted side AC across from the right angle. Using the theorem: 6² + 8² = c² → 36 + 64 = 100 → c = √100 = 10, so AC = 10 units. The distance formula gives the same result for A(1, 1) and C(7, 9): d = √[(7 − 1)² + (9 − 1)²] = √[36 + 64] = √100 = 10. Converse check: with sides 6, 8, 10 and 10 longest, 6² + 8² = 100 = 10², so the triangle is right.

3. Using Figure 1, suppose leg BC were 12 instead of 8 (with 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 (20–26 minutes) — Use what you know

4. Find a missing side.
   • 4a. Missing hypotenuse: legs a = 9 and b = 12; find c using a² + b² = c². Show every step.
   • 4b. Missing leg: one leg b = 5 and hypotenuse c = 13; find the other leg a. (The hypotenuse is known, so subtract: a² = c² − b².) Show your steps.
5. Distance between two plotted points: P(1, 2) and Q(7, 10). Find PQ with d = √[(x₂ − x₁)² + (y₂ − y₁)²]; show the horizontal change, the vertical change, both squares, the sum, and the square root.
6. Real-world application (ladder): a 17-foot ladder leans on a wall with its base 8 feet from the wall. How high up the wall does it reach? The ladder is the hypotenuse; the ground distance and wall height are the legs. Draw, label, and solve.
7. Error analysis: a student found the hypotenuse for legs a = 6 and b = 8 by writing "c = a + b = 6 + 8 = 14." Find the mistake, explain it, and give the correct hypotenuse using a² + b² = c².
8. Converse test: a triangle has sides 10, 24, and 26. Is it a right triangle? Identify the longest side, square all three, and check whether a² + b² = c². State your conclusion and why.

Block only (~12–15 minutes) — Extra applied task

9. Multi-step composite distance: a hiker's stops are R(0, 0), S(3, 4), and T(10, 4) on a grid (each unit = 1 km); she walks R to S, then S to T.
   • 9a. Find the length of segment RS with the distance formula. Show your work.
   • 9b. Find the length of segment ST with the distance formula. Show your work.
   • 9c. Find the total distance walked (RS + ST), then the straight-line distance from R to T. Which is longer, the path or the straight line? Explain in one sentence why that makes sense.

Explain (6–10 minutes) — Justify your strategy (Claim, Evidence, Reasoning)

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 it a right triangle? Decide and justify 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 "three different sides" 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 (5 minutes) — ACE

• Articulate: explain how to tell which side is the hypotenuse, and why you must take the square root at the end instead of stopping at c².
• Connect: name one packet item where the distance formula was really a² + b² = c², and name the legs.
• Extend: give a new real-life situation (not the ladder) where the Pythagorean Theorem finds a distance you cannot measure directly.

Continue (optional) — Early finisher

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; prove each works by showing a² + b² = c²; and explain in one sentence why scaling a triple always gives another right triangle. Then use the converse to show that 5-6-8 is not a right triangle.

Turn in

Hand in the whole packet with your name, class period, and date, with items 1–8, 10, and the ACE box answered and your work shown (item 9 too if you are on a block schedule), including exact and rounded answers where needed. Include the optional challenge if you did it.