Question

Find the measure of angles G and the length of GH

Answer 1

Since △GHI is a right triangle, then, from the Pythagorean Theorem, the lengths of its sides satisfy the following equation:

`GI^2=GH^2+HI^2`

Where GI is the hypotenuse of the right triangle, and GH and HI are the legs of the right triangle.

Since GH is unknown, isolate it from the equation:

`GH=\sqrt[]{GI^2-HI^2}`

Substitute the values of GI and HI:

`\begin{gathered} GH=\sqrt[]{26^2-10^2} \\ =\sqrt[]{676-100} \\ =\sqrt[]{576} \\ =24 \end{gathered}`

Therefore, GH=24.

On the other hand, recall the definition of sine of an angle on a right triangle:

`\sin (G)=\frac{\text{opposite leg}}{\text{hypotenuse}}`

The leg opposite to G is HI, and the hypotenuse of the right triangle is GI. Then:

`\begin{gathered} \sin (G)=(10)/(26) \\ \Rightarrow\sin (G)=(5)/(13) \end{gathered}`

Take the inverse sine function from both sides:

`\Rightarrow G=\sin ^(-1)((5)/(13))`

Use a calculator to find the inverse sine of 5/13:

`\therefore G=22.61986495\ldots`

Therefore, the angle G is approximately 22.6°.