The are of a triangle is:
[tex]A=\frac{1}{2}b\cdot h[/tex]
b is the base and h is the height
In the given triangle the base is Leg BC and the height is Leg AB:
[tex]\begin{gathered} AB=BC \\ A=12.5 \\ \\ 12.5=\frac{1}{2}AB^2 \end{gathered}[/tex]
Use the equation above to solve AB:
[tex]\begin{gathered} \text{Multiply both sides of the equation by 2:} \\ 12.5\times2=2\times\frac{1}{2}AB^2 \\ \\ 25=AB^2 \\ \\ \text{Take the square root of both sides of the equation:} \\ \sqrt[]{25}=\sqrt{AB^2} \\ \\ 5=AB \end{gathered}[/tex]
Then, Legs of the given right triangle have a measure of 5.
To find the measure of the hypotenuse use Pythagorean theorem:
[tex]\begin{gathered} \text{hypotenuse}^2=Leg1^2+Leg2^2 \\ \text{hypotenuse}=\sqrt[]{Leg1^2+Leg2^2} \end{gathered}[/tex][tex]\begin{gathered} \text{hypotenuse}=\sqrt[]{5^2+5^2} \\ \\ \text{hypotenuse}=\sqrt[]{25+25} \\ \\ \text{hypotenuse}=\sqrt[]{50} \end{gathered}[/tex]
Find the prime factorization of 50 to simplify it:
[tex]50=2\times5\times5=2\times5^2[/tex][tex]\begin{gathered} \text{hypotenuse}=\sqrt[]{2\times5^2} \\ \\ \text{hypotenuse}=5\sqrt[]{2} \end{gathered}[/tex]
