The Pythagorean Theorem and Its Converse
Build a square on each side of a right triangle and something neat happens: the two small squares fill exactly the big one. That fact is the Pythagorean Theorem, and running it backward tests for right angles.
Know these first
- Master List skill 8.EE.2
- Master List skill 6.G.1
This unlocks
What you'll be able to do
- Explain a² + b² = c² with the areas of squares built on the sides.
- Find the area or side of a missing square.
- Use the converse to decide whether three lengths make a right triangle.
- Test a triangle on a coordinate grid for a right angle.
- Tell acute, right and obtuse triangles apart from their side lengths.
The big idea
Build a square on each side of a right triangle. The two squares on the legs together have the same area as the square on the hypotenuse: a² + b² = c². For a triangle with sides 3, 4 and 5: 9 + 16 = 25.
Why it works: put four copies of the triangle inside a big square with side a + b, hypotenuses facing in. The hole in the middle is a square of area c². Rearrange the same four triangles into two rectangles, and the uncovered space becomes two squares, a² and b². Same big square, same four triangles, so the leftover areas are equal.
The converse runs the other way. If the two shorter sides of a triangle give a² + b² = c², where c is the longest side, the triangle has a right angle, across from c. If a² + b² is bigger than c², every angle is acute; if it's smaller, the triangle is obtuse.
Words you need
- Legs
- The two sides that form the right angle.
- Hypotenuse
- The longest side of a right triangle, across from the right angle.
- Converse
- A statement run backward: if a² + b² = c², then the triangle is right.
- Pythagorean triple
- Three whole numbers that fit a² + b² = c², like 3, 4, 5 or 5, 12, 13.
Worked examples
Example 1
A right triangle has squares built on its legs with areas of 36 and 64 square units. What is the area of the square on the hypotenuse?
- The square on the hypotenuse equals the two leg squares together.
- 36 + 64 = 100 square units. Its side, the hypotenuse, is √100 = 10.
Answer: 100
Example 2
Is a triangle with sides 9, 12 and 15 a right triangle?
- The longest side is 15, so test 9² + 12² against 15².
- 81 + 144 = 225 and 15² = 225.
- They match, so it's a right triangle, with the right angle across from the 15 side.
Answer: Yes, it is a right triangle
Example 3
Triangle ABC has vertices A(0, 0), B(4, 2) and C(3, 4). Is it a right triangle? If so, where is the right angle?
- Square each side length (no square roots needed): AB² = 4² + 2² = 20, BC² = 1² + 2² = 5, AC² = 3² + 4² = 25.
- The two shorter add to the longest: 20 + 5 = 25.
- So it's a right triangle. AC is the longest side, so the right angle is across from it, at B.
Answer: Yes - right angle at B
Common mistakes
Testing with the wrong side as c. c is always the longest side. Square the two shorter sides, add, and compare with the longest side squared.
Adding the sides instead of their squares. 3 + 4 isn't 5, but 9 + 16 is 25. The test uses squares, which are areas.
Deciding a triangle is right because it looks right. A picture can be off by a degree or two. Only the numbers decide.
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
- Squares are built on the legs of a right triangle. Their areas are 25 and 144. What is the area of the square on the hypotenuse?
- Is a triangle with sides 8, 15 and 17 a right triangle?
- A carpenter joins boards of 6, 9 and 11 feet into a triangle. Does the triangle have a square corner? Word problem
- Zoe tiles three square patches along the sides of a right-triangle flower bed. The patches on the two shorter sides cover 49 and 576 square feet. How much does the patch on the longest side cover? Word problem
- A right-angled shelf bracket has arms of 9 inches and 12 inches. How long is the brace that joins their ends? Word problem
Tier 2 Intermediate One more step to think through
- The square on the hypotenuse of a right triangle has an area of 225. The square on one leg has an area of 81. What is the area of the square on the other leg?
- Triangle ABC has vertices A(−1, 2), B(2, 5) and C(5, 2). Is it a right triangle? If so, where is the right angle?
- A sign-maker cuts a triangle with sides 20, 21 and 29 inches. Is it a right triangle? Word problem
- Two square tiles of 20 and 16 square inches fit exactly along the legs of a right triangle. How long is the hypotenuse? Word problem
- On a town map, a park is at A(0, 0), a school at B(6, 2) and a store at C(4, 8). Is the corner at the school a right angle? Word problem
- Lina wants a triangle whose sides of 10 cm and 24 cm meet at a right angle. How long must the third side be? Word problem
- Is a triangle with sides 1.5, 2 and 2.5 a right triangle?
Tier 3 Advanced Several steps - find the hidden structure
- A square frame has sides of 7 inches. Four right triangles with legs of 3 and 4 inches sit inside it, one in each corner, with their hypotenuses facing in. What is the area of the square hole in the middle? Word problem
- Triangle ABC has vertices A(0, 0), B(5, 1) and C(1, 4). Is it a right triangle?
- A builder marks 6 feet along one wall and 8 feet along the other, then measures 10.2 feet between the marks. Is the corner square? Word problem
- Triangle ABC has vertices A(1, 2), B(4, 6) and C(8, 3). Is it a right triangle? If so, where is the right angle?
- Three square rugs of 20, 45 and 65 square feet are laid edge to edge so they surround a triangle of floor. Is that triangle a right triangle? Word problem
Tier 4 Challenge Stretch problems, some above grade level
- A triangle has sides of 7, 8 and 12 inches. Is it acute, right or obtuse? Word problem Above grade level
- A triangle's longest side is 12 cm and another side is 9 cm. How long must the third side be for a right triangle? Round to the nearest tenth. Word problem
- Is a triangle with sides 10, 11 and 14 acute, right or obtuse? Above grade level
Answers
Tier 1 Foundations
- 169. The square on the hypotenuse equals the two leg squares together: 25 + 144 = 169.
- Yes, it is a right triangle. Check the longest side: 8² + 15² = 289 and 17² = 289, which match.
- No, it is not a right triangle. Check the longest side: 6² + 9² = 117 and 11² = 121, which do not match.
- 625. The square on the hypotenuse equals the two leg squares together: 49 + 576 = 625.
- 15. c² = 9² + 12² = 225, so c = 15.
Tier 2 Intermediate
- 144. The two leg squares add up to the hypotenuse square, so the missing one is 225 − 81 = 144.
- Yes - right angle at B. AB² = 18, BC² = 18, AC² = 36. The two shorter add to the longest (18 + 18 = 36), so it's a right triangle.
- Yes, it is a right triangle. Check the longest side: 20² + 21² = 841 and 29² = 841, which match.
- 6. The missing square's area is 20 + 16 = 36, so its side is √36 = 6.
- Yes - right angle at B. AB² = 40, BC² = 40, AC² = 80. The two shorter add to the longest (40 + 40 = 80), so it's a right triangle.
- 26. c² = 10² + 24² = 676, so c = 26.
- Yes, it is a right triangle. Check the longest side: 1.5² + 2² = 6.25 and 2.5² = 6.25, which match.
Tier 3 Advanced
- 25. The frame covers 7² = 49 square inches and each triangle covers 1/2 × 3 × 4 = 6, so the hole is 49 − 4 × 6 = 25. That's c² for c = 5: the proof in action.
- No - not a right triangle. AB² = 26, BC² = 25, AC² = 17. The two shorter give 25 + 17 = 42, not 26.
- No, it is not a right triangle. Check the longest side: 6² + 8² = 100 and 10.2² = 104.04, which do not match.
- 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.
- Yes, it is a right triangle. The two smaller squares add to 20 + 45 = 65, which equals the biggest square, 65.
Tier 4 Challenge
- Obtuse. 7² + 8² = 113 < 12² = 144, so the triangle is obtuse.
- 3√7 ≈ 7.9. b² = 12² − 9² = 63, so b = 3√7 ≈ 7.9.
- Acute. 10² + 11² = 221 > 14² = 196, so the triangle is acute.
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