Question

if (mangle fhg = mangle fhi = 39^{circ}), (fg = p + 9), and (fi = 10p), what is (fi?)

Explanation:

Step1: Apply angle - bisector theorem

Since $\angle FHG=\angle FHI = 39^{\circ}$, ray $HF$ is the angle - bisector of $\angle IHG$. By the angle - bisector theorem, in a triangle, if a ray bisects an angle of a triangle, then it divides the opposite side into segments that are proportional to the other two sides. In right - triangles $\triangle FHI$ and $\triangle FHG$, we have $FG = p + 9$ and $FI=10p$, and because of the angle - bisector property and the right - angles, $FG = FI$.

Step2: Set up the equation

Set $p + 9=10p$.

Step3: Solve for $p$

Subtract $p$ from both sides: $9 = 10p-p$. So, $9p=9$, then $p = 1$.

Step4: Find the value of $FI$

Substitute $p = 1$ into the expression for $FI$. Since $FI = 10p$, then $FI=10\times1=10$.

Answer:

$10$