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

We are given two lines representing equal sides of an isosceles triangle. The goal is to find the sum of all possible distinct values of the slope \( m \) of the third side.

To solve for the third side, we first calculate the intersection points of the given lines:

1. The first line is \( -x + 2y = 4 \), which can be rewritten as: \[ y = \frac{x + 4}{2} \]

2. The second line is \( x + y = 4 \), which simplifies to: \[ y = 4 - x \]

Now, we find the intersection of these two lines by solving the system of equations:

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

Solve this equation to find the point of intersection. Then, calculate the slopes of the lines formed by the points of intersection with the third side. Finally, sum all distinct possible slopes of the third side.

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