Question

which measurements could represent the side lengths in feet of a right triangle? a 10 ft, 24 ft, 26 ft b 2 ft, 3 ft, 5 ft c 14 ft, 14 ft, 14 ft d 3 ft, 3 ft, 18 ft

Explanation:

Step1: Recall Pythagorean theorem

For a right - triangle with side lengths \(a\), \(b\) (legs) and \(c\) (hypotenuse), \(a^{2}+b^{2}=c^{2}\), where \(c\) is the longest side.

Step2: Check option A

Let \(a = 10\), \(b = 24\) and \(c = 26\). Then \(a^{2}=10^{2}=100\), \(b^{2}=24^{2}=576\), and \(c^{2}=26^{2}=676\). Since \(a^{2}+b^{2}=100 + 576=676=c^{2}\), option A could represent the side - lengths of a right - triangle.

Step3: Check option B

Let \(a = 2\), \(b = 3\) and \(c = 5\). Then \(a^{2}=2^{2}=4\), \(b^{2}=3^{2}=9\), and \(c^{2}=5^{2}=25\). Since \(a^{2}+b^{2}=4 + 9 = 13
eq25=c^{2}\), option B does not represent the side - lengths of a right - triangle.

Step4: Check option C

The side - lengths \(14\), \(14\), \(14\) form an equilateral triangle. In an equilateral triangle, all angles are \(60^{\circ}\), so it is not a right - triangle.

Step5: Check option D

The sum of the lengths of two sides \(3+3 = 6<18\). According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. So, option D does not form a triangle at all.

Answer:

A. 10 ft, 24 ft, 26 ft