Question

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). square of length of the hypotenuse of the triangle = (square). (simplify your answer.)

Explanation:

Step1: Recall the Pythagorean theorem

The Pythagorean theorem states that for a right - triangle with legs of lengths \(a\) and \(b\) and hypotenuse of length \(c\), \(a^{2}+b^{2}=c^{2}\), where \(d(A,B)\) and \(d(B,C)\) can be considered as the lengths of the legs and \(d(A,C)\) as the length of the hypotenuse. Given \([d(A,B)]^{2}=53\), \([d(B,C)]^{2}=53\), and \([d(A,C)]^{2}=106\).

Step2: Calculate the sum of the squares of the legs

The sum of the squares of the lengths of the legs of the triangle is \([d(A,B)]^{2}+[d(B,C)]^{2}\). Substitute the given values: \(53 + 53=106\).

Step3: Find the square of the hypotenuse

The square of the length of the hypotenuse \(c^{2}=[d(A,C)]^{2}=106\). So, the length of the hypotenuse \(c = \sqrt{106}\).

Answer:

\(\sqrt{106}\)