Question

Two equal sides of an isosceles triangle are along \( -x + 2y = 4 \) and \( x + y = 4 \). If \( m \) is the slope of its third side, then the sum of all possible distinct values of \( m \) is:

• A. \( -2\sqrt{10} \)
• B. 12
• C. 6
• D. -6

Hint:

Use trigonometric identities and the relationship between slopes to solve geometry-related problems involving lines and angles.

Answer 1

The Correct Option is C

Given two lines forming equal sides of an isosceles triangle, determine the sum of all unique slopes of the third side.

To find the third side, first determine the intersection points of the provided lines:

1. Line 1: \( -x + 2y = 4 \), rewritten as \( y = \frac{x + 4}{2} \).

2. Line 2: \( x + y = 4 \), simplified to \( y = 4 - x \).

Equate the expressions for \( y \) to find the intersection point:

\[ \frac{x + 4}{2} = 4 - x \]

Solve this equation for \( x \) and \( y \). Then, calculate the slopes of the lines connecting the intersection point to potential vertices of the third side. Finally, sum all distinct possible slopes (\( m \)) of the third side.

Answer: The sum of all possible distinct values of \( m \) is \( \boxed{6} \).