Since △GHI is a right triangle, then, from the Pythagorean Theorem, the lengths of its sides satisfy the following equation:
[tex]GI^2=GH^2+HI^2[/tex]
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:
[tex]GH=\sqrt[]{GI^2-HI^2}[/tex]
Substitute the values of GI and HI:
[tex]\begin{gathered} GH=\sqrt[]{26^2-10^2} \\ =\sqrt[]{676-100} \\ =\sqrt[]{576} \\ =24 \end{gathered}[/tex]
Therefore, GH=24.
On the other hand, recall the definition of sine of an angle on a right triangle:
[tex]\sin (G)=\frac{\text{opposite leg}}{\text{hypotenuse}}[/tex]
The leg opposite to G is HI, and the hypotenuse of the right triangle is GI. Then:
[tex]\begin{gathered} \sin (G)=\frac{10}{26} \\ \Rightarrow\sin (G)=\frac{5}{13} \end{gathered}[/tex]
Take the inverse sine function from both sides:
[tex]\Rightarrow G=\sin ^{-1}(\frac{5}{13})[/tex]
Use a calculator to find the inverse sine of 5/13:
[tex]\therefore G=22.61986495\ldots[/tex]
Therefore, the angle G is approximately 22.6°.
