Question: Finding a side length given the perimeter and side lengths with variables The perimeter of the…

Finding a side length given the perimeter and side lengths with variables

The perimeter of the triangle below is 90 units. Find the length of side $\overline{BC}$.

Write your answer without variables.

[Diagram of a triangle with sides labeled as follows: side $AB = 5z - 3$, side $BC = z + 3$, side $AC = 4z$.]

$BC = \boxed{\phantom{0}}$

Solution

To solve the problem, we need to find the length of side \( BC \). Given: - Perimeter of the triangle: \( 90 \) units - \( AB = 5x - 3 \) - \( AC = 4x \) - \( BC = x + 3 \) First, write the equation for the perimeter: \[ AB + AC + BC = 90 \] Substitute the expressions for the sides: \[ (5x - 3) + 4x + (x + 3) = 90 \] Combine like terms: \[ 5x - 3 + 4x + x + 3 = 90 \] \[ 10x = 90 \] Solve for \( x \): \[ x = 9 \] Substitute \( x = 9 \) back into the expression for \( BC \): \[ BC = x + 3 = 9 + 3 = 12 \] Therefore, the length of side \( BC \) is \( 12 \) units.