Determine whether a triangle with the given side lengths is a right triangle Side lengths Right triangle Not a right triangle Not enough information 8, 15, 17 6, 7, 9 18, 32, 40 10, 24, 26
The Correct Answer and Explanation is:
To determine whether a triangle is a right triangle, we use the Pythagorean Theorem, which states that for a triangle with side lengths aaa, bbb, and ccc, where ccc is the longest side, the triangle is a right triangle if:a2+b2=c2a^2 + b^2 = c^2a2+b2=c2
Let us evaluate each set of side lengths:
1. 8, 15, 17
Here, 17 is the longest side.82+152=64+225=2898^2 + 15^2 = 64 + 225 = 28982+152=64+225=289172=28917^2 = 289172=289
Since both values are equal, this is a right triangle.
2. 6, 7, 9
Here, 9 is the longest side.62+72=36+49=856^2 + 7^2 = 36 + 49 = 8562+72=36+49=8592=819^2 = 8192=81
Since 85 ≠ 81, this is not a right triangle.
3. 18, 32, 40
Here, 40 is the longest side.182+322=324+1024=134818^2 + 32^2 = 324 + 1024 = 1348182+322=324+1024=1348402=160040^2 = 1600402=1600
Since 1348 ≠ 1600, this is not a right triangle.
4. 10, 24, 26
Here, 26 is the longest side.102+242=100+576=67610^2 + 24^2 = 100 + 576 = 676102+242=100+576=676262=67626^2 = 676262=676
Since both values are equal, this is a right triangle.
Summary:
- 8, 15, 17 → Right triangle
- 6, 7, 9 → Not a right triangle
- 18, 32, 40 → Not a right triangle
- 10, 24, 26 → Right triangle
The Pythagorean Theorem allows us to check for a right triangle by verifying whether the sum of the squares of the two shorter sides equals the square of the longest side. No additional information is needed when all three side lengths are provided.
