Question

in the diagram below of triangle bcd, e is the mid - point of bd and f is the mid - point of cd. if ef=-21 + 3x, and bc=-24 + 4x, what is the measure of bc?

Explanation:

Step1: Apply mid - segment theorem

In a triangle, the line segment joining the mid - points of two sides is half the length of the third side. So, $EF=\frac{1}{2}BC$.

Step2: Set up the equation

Substitute the given expressions for $EF$ and $BC$ into the equation $EF = \frac{1}{2}BC$. We get $- 21+3x=\frac{1}{2}(-24 + 4x)$.

Step3: Solve the equation

First, multiply both sides of the equation by 2 to get rid of the fraction: $2(-21 + 3x)=-24 + 4x$.
Expand the left - hand side: $-42+6x=-24 + 4x$.
Subtract $4x$ from both sides: $-42+6x-4x=-24 + 4x-4x$, which simplifies to $-42 + 2x=-24$.
Add 42 to both sides: $-42+2x + 42=-24 + 42$, so $2x = 18$.
Divide both sides by 2: $x = 9$.

Step4: Find the measure of $BC$

Substitute $x = 9$ into the expression for $BC$. $BC=-24 + 4x=-24+4\times9=-24 + 36=12$.

Answer:

12