Question

1. if △def is an isosceles triangle, ∠d is the vertex angle, de = 4x - 7, ef = 3x + 2 and df = 6x - 15, find the value of x and the measure of each side.

Explanation:

Step1: Set equal the lengths of the congruent sides

Since $\triangle DEF$ is isosceles with $\angle D$ as the vertex - angle, the legs are congruent, so $DE = EF$.
$4x−7 = 3x + 2$

Step2: Solve the equation for $x$

Subtract $3x$ from both sides: $4x-3x−7=3x - 3x+2$, which simplifies to $x−7 = 2$.
Then add 7 to both sides: $x−7 + 7=2 + 7$, so $x=9$.

Step3: Find the length of each side

For $DE$: Substitute $x = 9$ into the expression for $DE$, $DE=4x−7=4\times9−7=36 - 7=29$.
For $EF$: Substitute $x = 9$ into the expression for $EF$, $EF=3x + 2=3\times9+2=27 + 2=29$.
For $DF$: Substitute $x = 9$ into the expression for $DF$, $DF=6x−15=6\times9−15=54 - 15=39$.

Answer:

$x = 9$, $DE=29$, $EF = 29$, $DF=39$