I was doing my math homework when I got a question that asked as follows: The area of a square drawn on the hypotenuse of a isosceles triangle is 24cm2. Find the lengths of the other two sides, $b$ and $c$. Even though you only have one number given, since you know it’s an isosceles triangle, you can find the two missing lengths.
I was wondering how far I can take this, so I came up with the question: The area of a square on the hypotenuse is 24cm2 let’s call this side $a$, the length of side $b$ is unknown however we know the area of side $c$ is $24 - b^2$. Is this question possible? I don’t think it is possible. All I know for certain is that $b$ and $c$ have to be both less than the square root of 24cm2. So, using the Pythagorean theorem we know for sure it could be any two real numbers squared that are less than 4.898979486 and have to add up to exactly that number while adding up to 24cm2 when squared.
Let’s call the lengths of the two sides $b$ and $c$.
Since it’s a right triangle, Pythagoras tells us that $b^2 + c^2 = 24$.
Since it’s an isosceles triangle, we know that $b=c$.
Can you take it from there?
If we don’t require the triangle to be isosceles, there are an infinite number of solutions. The triangle vertex that has a right angle could lie anywhere on a circle. See if you can figure out what circle.