Question

△def is an isosceles triangle with a perimeter of 74.
overline{eg} is a median of the triangle.
what is the length of overline{ef}?
a 7
b 9
c 28
d 18

Explanation:

Step1: Recall property of isosceles - triangle median

In an isosceles triangle, the median to the base is also the perpendicular - bisector. So, $FG = GD=x + 2$. Then $FD=2(x + 2)$.

Step2: Set up perimeter equation

The perimeter of $\triangle DEF$ is $P=EF + ED+FD$. Since it's isosceles, assume $EF = ED = 4x$. Then $P=4x+4x + 2(x + 2)$. Given $P = 74$, we have the equation $4x+4x+2(x + 2)=74$.

Step3: Simplify the equation

Expand the left - hand side: $4x+4x+2x + 4 = 74$. Combine like terms: $10x+4 = 74$.

Step4: Solve for $x$

Subtract 4 from both sides: $10x=74 - 4=70$. Divide both sides by 10: $x = 7$.

Step5: Find the length of $EF$

Since $EF = 4x$, substitute $x = 7$ into the expression. So, $EF=4\times7 = 28$.

Answer:

C. 28