Question

1. if △dgh ~ △def, find the value of x. x = 25

Explanation:

Step1: Use similarity - side ratio property

Since $\triangle DGH\sim\triangle DEF$, the ratios of corresponding sides are equal. We have $\frac{DG}{DE}=\frac{DH}{DF}=\frac{GH}{EF}$. Looking at the sides with expressions involving $x$, we consider $\frac{GH}{EF}=\frac{DG}{DE}$. The lengths $DG = 52$, $DE=91$, $GH=x + 3$, and $EF=2x - 1$. So, $\frac{x + 3}{2x - 1}=\frac{52}{91}$.

Step2: Cross - multiply

Cross - multiplying the proportion $\frac{x + 3}{2x - 1}=\frac{52}{91}$ gives $91(x + 3)=52(2x - 1)$.

Step3: Expand both sides

Expand $91(x + 3)$ to get $91x+273$, and expand $52(2x - 1)$ to get $104x-52$. So, $91x + 273=104x-52$.

Step4: Solve for $x$

First, move the $x$ terms to one side and constants to the other side: $273 + 52=104x-91x$. This simplifies to $325 = 13x$. Then, divide both sides by 13: $x=\frac{325}{13}=25$.

Answer:

$x = 25$