Question

to show that the triangle is a right triangle, show that the sum of the squares
find the length of each line segment.
d(a,b) = \sqrt{53}
d(a,c) = \sqrt{106}
d(b,c) = \sqrt{53}
(simplify your answers. type exact answers, using radicals as needed.)
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 = 106
square of length of the hypotenuse of the triangle = 106
(simplify your answers.)
find the area of the triangle.
area = square units

Explanation:

Step1: Identify base and height

Since it's a right - triangle and \(d(A,B)=\sqrt{53}\), \(d(B,C)=\sqrt{53}\), we can consider these as the legs (base and height).

Step2: Apply area formula for right - triangle

The area formula for a right - triangle is \(A=\frac{1}{2}bh\). Here \(b = \sqrt{53}\) and \(h=\sqrt{53}\), so \(A=\frac{1}{2}\times\sqrt{53}\times\sqrt{53}\).

Step3: Simplify the expression

\(A=\frac{1}{2}\times53=\frac{53}{2}\).

Answer:

\(\frac{53}{2}\)