Question

mathematics progress check
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.
vertex a (12,-8)
vertex b (-10,-15)
vertex c (-8,14)
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. yes. the triangle is an isosceles triangle but not a right triangle.
b. no. the triangle is both an isosceles triangle and a right triangle.
c. no. the triangle is a right triangle but not an isosceles triangle.
d. no. the triangle is neither an isosceles triangle nor 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 \(AB\): \(x_1 = 12,y_1=-8,x_2=-10,y_2=-16\)
\(AB=\sqrt{(-10 - 12)^2+(-16+8)^2}=\sqrt{(-22)^2+(-8)^2}=\sqrt{484 + 64}=\sqrt{548}\)
For \(BC\): \(x_1=-10,y_1=-16,x_2=-8,y_2 = 14\)
\(BC=\sqrt{(-8 + 10)^2+(14 + 16)^2}=\sqrt{(2)^2+(30)^2}=\sqrt{4+900}=\sqrt{904}\)
For \(AC\): \(x_1 = 12,y_1=-8,x_2=-8,y_2 = 14\)
\(AC=\sqrt{(-8 - 12)^2+(14 + 8)^2}=\sqrt{(-20)^2+(22)^2}=\sqrt{400+484}=\sqrt{884}\)

Step2: Check for isosceles triangle

Since \(AB=\sqrt{548}\), \(BC=\sqrt{904}\), \(AC=\sqrt{884}\), no two sides are equal. So it is not an isosceles triangle.

Step3: Check for right - triangle

Use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\)
\(AB^{2}=548\), \(BC^{2}=904\), \(AC^{2}=884\)
\(AB^{2}+AC^{2}=548 + 884=1432
eq904=BC^{2}\)
\(AB^{2}+BC^{2}=548+904 = 1452
eq884=AC^{2}\)
\(AC^{2}+BC^{2}=884+904=1788
eq548=AB^{2}\)

Answer:

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