Question

1. ABC is an Isosceles-Right Triangle. AB is tangent to the circle at D. If mAH = 3, mHG = 9, and mGC=2, find: a) mEF b) mBE c) mFC​

Answer 1

Step-by-step explanation:

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Answer 2

Given:

mAH = 3

mHG = 9

mGC=2

$m \overline{A C}=m \overline{A H}+m \overline{H G}+m \overline{G C}$

$\begin{array}{l}=3+9+2 \\=14\end{array}$

$\begin{aligned}A B^{2}+B C^{2} &=A C^{2} \\A B=B C & \\\therefore 2 B C^{2} &=A C^{2} \\B C^{2} &=98 .\end{aligned}$

$B C=4 \sqrt{2}$

$\begin{array}{l}m \overline{A H}: m \overline{H G}: m \overline{G C}=3: 9: 2 \\B E: E F: F C=3: 9: 2 . \\\therefore \text { Set } m \overline{E F}=9 x, \quad m \overline{B I}=3 x, m \overline{F l}=2 x\end{array}$

$\begin{array}{c}14 x=7 \sqrt{2} \\x=\frac{\sqrt{2}}{2}\end{array}$

$\text { a) } m \overline{F L}=9 x\\\\=\frac{9 \sqrt{2}}{2}$

$\text { b) } m \overline{B F}=3 x\\\\=\frac{3 \sqrt{2}}{2}$

$\text { c) } \quad m \overline{F C}=2 x\\\\=\sqrt{2}$