Question

on each side of the bench swing, the chains which support the bench form a triangle. brian wanted to know what type of triangles were formed. he assigned coordinates to represent the vertices of the triangle formed to help him classify the triangle

after comparing the slopes and lengths of the three sides of the triangle, brian concluded the triangle formed is an isosceles triangle but not a right triangle.
did brian correctly classify the triangle?
a. no. the triangle is neither an isosceles triangle nor a right triangle.
b. no. the triangle is a right triangle but not an isosceles triangle.
c. no. the triangle is both an isosceles triangle and a right triangle.
d. yes. the triangle is an isosceles triangle but not a right triangle.

Explanation:

Step1: Calculate the lengths of the sides

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)
For \(AB\):
\(x_1 = 12,y_1=-6,x_2=-10,y_2 = 10\)
\(AB=\sqrt{(-10 - 12)^2+(10+6)^2}=\sqrt{(-22)^2+16^2}=\sqrt{484 + 256}=\sqrt{740}\)
For \(BC\):
\(x_1=-10,y_1 = 10,x_2=-3,y_2 = 14\)
\(BC=\sqrt{(-3 + 10)^2+(14 - 10)^2}=\sqrt{7^2+4^2}=\sqrt{49+16}=\sqrt{65}\)
For \(AC\):
\(x_1 = 12,y_1=-6,x_2=-3,y_2 = 14\)
\(AC=\sqrt{(-3 - 12)^2+(14 + 6)^2}=\sqrt{(-15)^2+20^2}=\sqrt{225+400}=\sqrt{625}=25\)

Step2: Check for right - triangle

Use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\)
\(BC^{2}+AB^{2}=65 + 740=805
eq625\), \(AB^{2}+AC^{2}=740+625 = 1365
eq65\), \(BC^{2}+AC^{2}=65+625=690
eq740\)

Step3: Check for isosceles triangle

Since no two sides (\(AB=\sqrt{740}\approx27.2\), \(BC=\sqrt{65}\approx8.06\), \(AC = 25\)) are equal.

Answer:

A. No. The triangle is neither an isosceles triangle nor a right triangle.