Question

The area of a right angle is 50. One of its angles is 45 degrees. Find the lengths of the sides and hypotenuse of the triangle.

Answer 1

It's isosceles triangle (look at the picture).

The area of the this triangle: $A=\dfrac{a^2}{2}$

We have $A=50$

Substitute:

$\dfrac{a^2}{2}=50\ \ \ |\cdot2\\\\a^2=100\to a=\sqrt{100}\to a=10$

Use the Pythagorean theorem:

$b^2=a^2+a^2\\\\b^2=100+100\\\\b^2=200\to b=\sqrt{200}\\\\b=\sqrt{100\cdot2}\\\\b=\sqrt{100}\cdot\sqrt2\\\\b=10\sqrt2$

Answer: 10, 10, 10√2