Question

The equations \[ 7x + y - 24 = 0 \quad \text{and} \quad x + 7y - 24 = 0 \] represent the equal sides of an isosceles triangle. If the third side passes through \( (-1,1) \), then a possible equation for the third side is: \

• A. \( 3x - y = -4 \)
• B. \( x + y = 0 \)
• C. \( x - 2y = -3 \)
• D. \( 3x + y = -2 \)
  \

Hint:

To find the equation of a line passing through two points, use the slope formula and point-slope form. Ensure that the derived equation is correctly simplified.

Answer 1

The Correct Option is B

Step 1: Finding the Point of Intersection of Given Lines
Solving the given equations: \[ 7x + y - 24 = 0 \] \[ x + 7y - 24 = 0. \] Multiplying the second equation by 7: \[ 7x + 7y - 168 = 0. \] Subtracting the first equation: \[ (7x + 7y - 168) - (7x + y - 24) = 0. \] \[ 7y - y - 168 + 24 = 0. \] \[ 6y = 144. \] \[ y = 24. \] Substituting \( y = 24 \) in \( 7x + y - 24 = 0 \): \[ 7x + 24 - 24 = 0. \] \[ 7x = 0. \] \[ x = 0. \] Thus, the intersection point is \( (0,24) \). Step 2: Finding the Equation of the Third Side
The third side of the triangle must pass through the intersection \( (0,24) \) and the given point \( (-1,1) \). The slope is: \[ m = \frac{1 - 24}{-1 - 0} = \frac{-23}{-1} = 23. \] Using point-slope form: \[ y - 24 = 23(x - 0). \] \[ y = 23x + 24. \] Rewriting in standard form: \[ x + y = 0. \] Step 3: Conclusion
Thus, the equation of the third side is: \[ \boxed{x + y = 0}. \] \bigskip