📐 Geometry · Similarity and Right Triangles
The Pythagorean Theorem: true or false
Decide fast, then read the reason. Wrong answers are the best teachers, so every statement explains itself.
All statements and answers
In every right triangle, the hypotenuse is the longest side.
✓ True. Since c2 = a2 + b2 is larger than a2 and larger than b2, the hypotenuse c is longer than each leg.
A right triangle with legs 6 and 11 has a hypotenuse of 17.
✗ False. That adds the legs. Square first: 36 + 121 = 157, so c = √157 ≈ 12.53. The hypotenuse must be shorter than 6 + 11 = 17.
A right triangle with legs 18 and 24 has a hypotenuse of 30.
✓ True. 182 + 242 = 324 + 576 = 900 = 302. It is the 3-4-5 triple multiplied by 6.
A triangle with side lengths 4, 5 and 6 is a right triangle.
✗ False. The longest side is 6, so test 42 + 52 = 41 against 62 = 36. They are not equal.
A triangle with side lengths 85, 13 and 84 is a right triangle.
✓ True. The longest side is 85. 132 + 842 = 169 + 7056 = 7225, and 852 = 7225.
For every triangle with sides a, b and c, a2 + b2 = c2.
✗ False. The theorem holds only for right triangles. An equilateral triangle with all sides 2 gives 4 + 4 = 8, but c2 = 4.
A right triangle with legs 2 and 9 has a hypotenuse of 85.
✗ False. The sum 22 + 92 = 85 is c2. The hypotenuse is c = √85 ≈ 9.22.
If three side lengths satisfy a2 + b2 = c2 with c the longest side, the triangle is a right triangle.
✓ True. This is the converse of the Pythagorean theorem, and it is true.
To find a missing leg, add the squares of the other two sides.
✗ False. Add only when the hypotenuse is the unknown. For a missing leg, subtract the known leg’s square from the hypotenuse’s square.
√72 = 6√5.
✗ False. Since 72 = 36 × 2, √72 = 6√2. Check: (6√2)2 = 36 × 2 = 72, but (6√5)2 = 36 × 5 = 180.
√20 = 4.47.
✗ False. The decimal is rounded, so write √20 ≈ 4.47. (Also √20 = 2√5 in simplified form.)
Multiplying all three numbers of a Pythagorean triple by the same positive whole number gives another Pythagorean triple.
✓ True. (ka)2 + (kb)2 = k2(a2 + b2) = k2c2 = (kc)2.
If you know any two sides of a right triangle, you can find the third side.
✓ True. Two legs give the hypotenuse by adding their squares. A hypotenuse and a leg give the other leg by subtracting.
Every right triangle has three whole-number side lengths.
✗ False. Only Pythagorean triples do. A right triangle with legs 1 and 2 has hypotenuse √5 ≈ 2.24.