Question

to show that the triangle is a right triangle, show that the sum of the squares of the lengths of two of the sides (the legs) equals the square of the length of the third side (the hypotenuse). find the length of each line - segment. d(a,b)=\sqrt{53} d(a,c)=\sqrt{106} d(b,c)=\sqrt{53} find the squared distance of each segment. d(a,b)^2 = 53 d(a,c)^2 = 106 d(b,c)^2 = 53 simplify your answers. type exact answers, using radicals as needed.) sum of the squares of the lengths of the legs of the triangle = square of length of the hypotenuse of the triangle = (simplify your answers.)

Explanation:

Step1: Recall Pythagorean theorem

For a right - triangle, if the lengths of the legs are \(a\) and \(b\), and the length of the hypotenuse is \(c\), then \(a^{2}+b^{2}=c^{2}\). Here, we are given \(d(A,B)=\sqrt{53}\), \(d(A,C)=\sqrt{106}\), \(d(B,C)=\sqrt{53}\). The squares of the lengths are \([d(A,B)]^{2}=53\), \([d(A,C)]^{2}=106\), \([d(B,C)]^{2}=53\).

Step2: Check Pythagorean theorem

We need to check if the sum of the squares of the lengths of two sides equals the square of the length of the third side. Notice that \([d(A,B)]^{2}+[d(B,C)]^{2}=53 + 53=106=[d(A,C)]^{2}\).

Answer:

The triangle is a right - triangle. The length of the hypotenuse is \(\sqrt{106}\) and the lengths of the legs are \(\sqrt{53}\) and \(\sqrt{53}\).