Question

1. use the distance formula to show that \\( \triangle abc \\) is isosceles. 14. draw a scalene triangle on a coordinate plane, and use the distance formula to demonstrate that your triangle is scalene.

Explanation:

Step1: Find coordinates of points

From the graph, \(A(-3,-1)\), \(B(-1,4)\), \(C(1,-1)\)

Step2: Apply distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

For \(AB\):
\(x_1=-3,y_1 = - 1,x_2=-1,y_2 = 4\)
\(AB=\sqrt{(-1+3)^2+(4 + 1)^2}=\sqrt{4 + 25}=\sqrt{29}\)
For \(BC\):
\(x_1=-1,y_1 = 4,x_2=1,y_2=-1\)
\(BC=\sqrt{(1 + 1)^2+(-1 - 4)^2}=\sqrt{4+25}=\sqrt{29}\)
For \(AC\):
\(x_1=-3,y_1=-1,x_2=1,y_2=-1\)
\(AC=\sqrt{(1 + 3)^2+(-1+1)^2}=\sqrt{16+0}=4\)

Answer:

Since \(AB = BC=\sqrt{29}\), \(\triangle ABC\) is isosceles.