Question

the vertices of a triangle are listed below. e(5,4), f(10,2), g(5,-2) which of the following correctly classifies the triangle? a. the triangle is a right scalene triangle. b. the triangle is an acute equilateral triangle. c. the triangle is an acute scalene triangle. d. the triangle is an acute isosceles triangle.

Explanation:

Step1: Calculate the lengths of the sides using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

For side \(EF\): \(x_1 = 5,y_1 = 4,x_2 = 10,y_2 = 2\)
\(EF=\sqrt{(10 - 5)^2+(2 - 4)^2}=\sqrt{25 + 4}=\sqrt{29}\)
For side \(FG\): \(x_1 = 10,y_1 = 2,x_2 = 5,y_2=-2\)
\(FG=\sqrt{(5 - 10)^2+(-2 - 2)^2}=\sqrt{25 + 16}=\sqrt{41}\)
For side \(EG\): \(x_1 = 5,y_1 = 4,x_2 = 5,y_2=-2\)
\(EG=\sqrt{(5 - 5)^2+(-2 - 4)^2}=\sqrt{0 + 36}=6\)

Step2: Check the type of triangle

Since \(EG = 6\), and \(EF=\sqrt{29}\approx5.39\), \(FG=\sqrt{41}\approx6.40\), two sides (\(EF\) and \(FG\)) are not equal, so it's not isosceles or equilateral.
Also, check if it's a right - triangle using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\)
\(EF^{2}+EG^{2}=29 + 36=65
eq41=FG^{2}\), \(EF^{2}+FG^{2}=29+41 = 70
eq36=EG^{2}\), \(EG^{2}+FG^{2}=36 + 41=77
eq29=EF^{2}\)
All angles are acute (because \(a^{2}+b^{2}>c^{2}\) for all combinations of sides \(a,b,c\)) and all sides are of different lengths.

Answer:

C. The triangle is an acute scalene triangle