Question

to show that the triangle is a right triangle, show that the sum of the squares of the lengths of
find the length of each line segment.
d(a,b)= 10
d(a,c)= 3\sqrt{10}
d(b,c)= \sqrt{10}
(simplify your answers. type exact answers, using radicals as needed.)
find the squared distance of each segment.
d(a,b)^2 = 100
d(a,c)^2 = 90
d(b,c)^2 = 10
(simplify your answers. type exact answers, using radicals as needed.)
the sum of the squares of the lengths of the legs of the triangle is .
the square of the length of the hypotenuse of the triangle is .
(simplify your answers.)

Explanation:

Step1: Identify the legs and hypotenuse

In a right - triangle, the hypotenuse is the longest side. Here, $d(A,C)=3\sqrt{10}$ is the longest side, and $d(A,B) = 10$ and $d(B,C)=\sqrt{10}$ are the legs.

Step2: Calculate the sum of the squares of the legs

$[d(A,B)]^{2}+[d(B,C)]^{2}=100 + 10=110$

Step3: Calculate the square of the hypotenuse

$[d(A,C)]^{2}=90$

Answer:

The sum of the squares of the lengths of the legs of the triangle is $110$.
The square of the length of the hypotenuse of the triangle is $90$.