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:
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:
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Use trigonometric identities and the relationship between slopes to solve geometry-related problems involving lines and angles.
Updated On: Mar 17, 2025
- \( -2\sqrt{10} \)
- 12
- 6
- -6
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The Correct Option is C
Solution and Explanation
Step 1: Derive the Equation for the Third Side Slope
The equation for the slope of the third side is given by the angle between two lines formula: \[ \tan(\theta) = \frac{m - \frac{1}{2}}{1 + \frac{1}{2}m} \]
Step 2: Solve the Resulting Quadratic EquationRearranging and simplifying the terms, we obtain a quadratic equation: \[ 2m^2 - 3m + 1 = m^2 + 3m + 2 \]
Step 3: Calculate the Sum of the RootsSolving the simplified equation leads to: \[ m_1 + m_2 = 6 \]
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