Question

In right Delta*A * B * C legAB = legBC The area of the triangle is 12.5. Hypotenuse AC equals to : A. sqrt(5) B. 5sqrt(2) C. 5 D.4 sqrt 5

Answer 1

The are of a triangle is:

`A=(1)/(2)b\cdot h`

b is the base and h is the height

In the given triangle the base is Leg BC and the height is Leg AB:

`\begin{gathered} AB=BC \\ A=12.5 \\ \\ 12.5=(1)/(2)AB^2 \end{gathered}`

Use the equation above to solve AB:

`\begin{gathered} \text{Multiply both sides of the equation by 2:} \\ 12.5*2=2*(1)/(2)AB^2 \\ \\ 25=AB^2 \\ \\ \text{Take the square root of both sides of the equation:} \\ \sqrt[]{25}=√(AB^2) \\ \\ 5=AB \end{gathered}`

Then, Legs of the given right triangle have a measure of 5.

To find the measure of the hypotenuse use Pythagorean theorem:

`\begin{gathered} \text{hypotenuse}^2=Leg1^2+Leg2^2 \\ \text{hypotenuse}=\sqrt[]{Leg1^2+Leg2^2} \end{gathered}`
`\begin{gathered} \text{hypotenuse}=\sqrt[]{5^2+5^2} \\ \\ \text{hypotenuse}=\sqrt[]{25+25} \\ \\ \text{hypotenuse}=\sqrt[]{50} \end{gathered}`

Find the prime factorization of 50 to simplify it:

`50=2*5*5=2*5^2`
`\begin{gathered} \text{hypotenuse}=\sqrt[]{2*5^2} \\ \\ \text{hypotenuse}=5\sqrt[]{2} \end{gathered}`