Question

in the diagram below, ∠fde≅∠fgh, gd = 1.5, he = 2.6, and fg = 2.5. find the length of fh. round your answer to the nearest tenth if necessary.

Explanation:

Step1: Identify similar triangles

Since $\angle FDE\cong\angle FGH$, and $\angle F$ is common to both $\triangle FDE$ and $\triangle FGH$, by the AA (angle - angle) similarity criterion, $\triangle FGH\sim\triangle FDE$.

Step2: Set up proportion

For similar triangles $\triangle FGH$ and $\triangle FDE$, we have the proportion $\frac{FG}{FD}=\frac{FH}{FE}$. Let $FD = FG + GD=2.5 + 1.5=4$, and $FE=FH + HE=FH + 2.6$. So $\frac{2.5}{4}=\frac{FH}{FH + 2.6}$.

Step3: Cross - multiply

Cross - multiplying the proportion $\frac{2.5}{4}=\frac{FH}{FH + 2.6}$ gives $2.5(FH + 2.6)=4FH$.
Expanding the left - hand side: $2.5FH+6.5 = 4FH$.

Step4: Solve for FH

Subtract $2.5FH$ from both sides: $6.5=4FH - 2.5FH$.
Combining like terms: $6.5 = 1.5FH$.
Then $FH=\frac{6.5}{1.5}=\frac{65}{15}=\frac{13}{3}\approx4.3$.

Answer:

$4.3$