Question

the points d(4, -2), e(-5, -2), and f(1, -8) form a triangle. graph the triangle on the set of axes below and find the perimeter in simplest radical form.

Explanation:

Step1: Calculate the length of \( DE \)

Since \( D(4,-2) \) and \( E(-5,-2) \), using the distance formula \( d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} \). Here \( y_1=y_2=-2 \), so \( DE=\vert4-(-5)\vert=\vert4 + 5\vert=9 \)

Step2: Calculate the length of \( DF \)

For \( D(4,-2) \) and \( F(1,-8) \), \( d=\sqrt{(1 - 4)^2+(-8+2)^2}=\sqrt{(-3)^2+(-6)^2}=\sqrt{9 + 36}=\sqrt{45}=3\sqrt{5} \)

Step3: Calculate the length of \( EF \)

For \( E(-5,-2) \) and \( F(1,-8) \), \( d=\sqrt{(1+5)^2+(-8 + 2)^2}=\sqrt{6^2+(-6)^2}=\sqrt{36+36}=\sqrt{72}=6\sqrt{2} \)

Step4: Calculate the perimeter \( P \)

\( P=DE+DF+EF=9 + 3\sqrt{5}+6\sqrt{2} \)

Answer:

\( 9 + 3\sqrt{5}+6\sqrt{2} \)