Question

select the correct answer.
the points d(-2,2), e(2,5), and f(5,1) form △def in the coordinate plane. what condition verifies that the triangle is a right triangle?
a. the product of the slopes of two of the sides is -1.
b. the product of the slopes of two of the sides is 1.
c. three of the sides of the triangle are equal.
d. two of the sides of the triangle are equal.

Explanation:

Step1: Recall slope formula

The slope \( m \) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Calculate slopes of \( DE \), \( EF \), \( FD \)

• Slope of \( DE \): \( D(-2,2) \), \( E(2,5) \). \( m_{DE}=\frac{5 - 2}{2 - (-2)}=\frac{3}{4} \).
• Slope of \( EF \): \( E(2,5) \), \( F(5,1) \). \( m_{EF}=\frac{1 - 5}{5 - 2}=\frac{-4}{3} \).
• Slope of \( FD \): \( F(5,1) \), \( D(-2,2) \). \( m_{FD}=\frac{2 - 1}{-2 - 5}=\frac{1}{-7}=-\frac{1}{7} \).

Step3: Check product of slopes

Multiply \( m_{DE} \) and \( m_{EF} \): \( \frac{3}{4} \times \frac{-4}{3} = -1 \). When the product of slopes of two lines is -1, they are perpendicular (right angle).

Answer:

A. The product of the slopes of two of the sides is -1