Question

plot each point and form the right triangle abc. verify that the triangle is a right triangle. find its area. a = (-5,3); b = (4,0); c = (1,6) 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) = 3\sqrt{10} d(a,c) = 3\sqrt{5} d(b,c) = 3\sqrt{5} (simplify your answers. type exact answers, using radicals as needed.) find the squared distance of each segment. d(a,b)^2 = d(a,c)^2 = d(b,c)^2 = (simplify your answers. type exact answers, using radicals as needed.)

Explanation:

Step1: Square the given distances

We know that if \(d(A,B)=3\sqrt{10}\), \(d(A,C) = 3\sqrt{5}\), and \(d(B,C)=3\sqrt{5}\), then \([d(A,B)]^{2}=(3\sqrt{10})^{2}=9\times10 = 90\), \([d(A,C)]^{2}=(3\sqrt{5})^{2}=9\times5=45\), \([d(B,C)]^{2}=(3\sqrt{5})^{2}=9\times5 = 45\).

Step2: Check the Pythagorean - theorem

The Pythagorean theorem states that for a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\). Here, \([d(A,C)]^{2}+[d(B,C)]^{2}=45 + 45=90=[d(A,B)]^{2}\), so the triangle is a right - triangle.

Step3: Calculate the area of the right - triangle

The area of a right - triangle is \(A=\frac{1}{2}\times\text{base}\times\text{height}\). The legs of the right - triangle are \(d(A,C)\) and \(d(B,C)\). So \(A=\frac{1}{2}\times3\sqrt{5}\times3\sqrt{5}=\frac{1}{2}\times9\times5=\frac{45}{2}\).

Answer:

\([d(A,B)]^{2}=90\), \([d(A,C)]^{2}=45\), \([d(B,C)]^{2}=45\), and the area of the right - triangle is \(\frac{45}{2}\)