Question

directions: find the measures of the sides of \\( \triangle def \\), then classify it by its sides.

1. \\( d(8,-6), e(-1,-3), f(-2,5) \\)

Explanation:

Step1: Use the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For side \(DE\):
\(x_1 = 8,y_1=-6,x_2=-1,y_2 = -3\)
\(DE=\sqrt{(-1 - 8)^2+(-3+6)^2}=\sqrt{(-9)^2 + 3^2}=\sqrt{81 + 9}=\sqrt{90}=3\sqrt{10}\)

Step2: For side \(EF\)

\(x_1=-1,y_1=-3,x_2=-2,y_2 = 5\)
\(EF=\sqrt{(-2 + 1)^2+(5 + 3)^2}=\sqrt{(-1)^2+8^2}=\sqrt{1 + 64}=\sqrt{65}\)

Step3: For side \(DF\)

\(x_1 = 8,y_1=-6,x_2=-2,y_2 = 5\)
\(DF=\sqrt{(-2 - 8)^2+(5 + 6)^2}=\sqrt{(-10)^2+11^2}=\sqrt{100+121}=\sqrt{221}\)

Answer:

Since \(DE = 3\sqrt{10}\), \(EF=\sqrt{65}\), \(DF=\sqrt{221}\) and \(DE
eq EF
eq DF\), \(\triangle DEF\) is a scalene triangle.