Question

The equal sides of an isosceles triangle are given by equations \( 7x - y + 3 = 0 \) and \( x + y - 3 = 0 \). If the slope \( m \) of the third side is an integer, then \( m = \):

• A. \( -3 \)
• B. \( 3 \)
• C. \( 4 \)
• D. \( -1 \)

Hint:

When solving for the equation of a line and finding the slope, be sure to carefully simplify and substitute values into the slope formula.

Answer 1

The Correct Option is A

We are given the equations of the two equal sides of the isosceles triangle: 1. \( 7x - y + 3 = 0 \) 2. \( x + y - 3 = 0 \)
First, we solve these two equations to find the coordinates of the vertex of the isosceles triangle. From equation 2, we solve for \( y \): \[ y = 3 - x \] Substitute this into equation 1: \[ 7x - (3 - x) + 3 = 0 \] Simplifying: \[ 7x - 3 + x + 3 = 0 \\ 8x = 0 \\ x = 0 \] Now, substitute \( x = 0 \) into \( y = 3 - x \): \[ y = 3 \] Thus, the vertex of the triangle is at \( (0, 3) \). Now, we find the slope of the third side. The third side passes through the point \( (0, 3) \) and is connected to the point on the x-axis, which is \( (3, 0) \) (from the second equation). The slope \( m \) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substitute the points \( (x_1, y_1) = (0, 3) \) and \( (x_2, y_2) = (3, 0) \): \[ m = \frac{0 - 3}{3 - 0} = \frac{-3}{3} = -1 \] Thus, the slope of the third side is \( m = -3 \).