Question

a right triangle has side lengths 28 centimeters, 45 centimeters, and 53 centimeters. what are the lengths of the legs and why? 45 and 53 centimeters, because they are the two longest sides. 45 and 53 centimeters, because 28² + 45² = 53². 28 and 45 centimeters, because 28 and 45 are both composite numbers. 28 and 45 centimeters, because they are the two shortest sides.

Explanation:

Step1: Recall the Pythagorean theorem

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

Step2: Check the sum of squares

Calculate \(28^{2}=28\times28 = 784\), \(45^{2}=45\times45=2025\), and \(53^{2}=53\times53 = 2809\). Then \(28^{2}+45^{2}=784 + 2025=2809=53^{2}\). Also, the legs of a right - triangle are the two shorter sides. Since \(28<45<53\), the two shorter sides are \(28\) and \(45\).

Answer:

The lengths of the legs are \(28\) and \(45\) centimeters. The first option is wrong because the legs are not the two longest sides. The second option is wrong in the identification of legs (it should be the two shorter sides that satisfy \(a^{2}+b^{2}=c^{2}\)). The third option is wrong because being a composite number has no relation to being a leg of a right - triangle. The correct answer is \(28\) and \(45\) centimeters, because they are the two shortest sides (and also satisfy \(28^{2}+45^{2}=53^{2}\)).