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

The equation for the slope of the third side is derived from the angle between two lines, given by the formula: \[ \tan(\theta) = \frac{m - \frac{1}{2}}{1 + \frac{1}{2}m} \] Solving the equation for the third side slope, we get a quadratic equation for \( m \): \[ 2m^2 - 3m + 1 = m^2 + 3m + 2 \] Simplifying and solving for the sum of the roots, we get \( m_1 + m_2 = 6 \).