In a certain isosceles right triangle, the altitude to the hypotenuse has length 4#sqrt(2)# . What is the area of the triangle?
Any isosceles right triangle is half a square, cut by its diagonal.
So, the altitude to the hypotenuse is half the diagonal of the square (which also means that the altitude to the hypotenuse is half the hypotenuse, by the way).
This means that the diagonal of the square is
In any square, you have
So, solving for
In this case, this means that the side of the square is
This, in turn, means that the area of the square is
Since the triangle is half the square, its area will be
Since the altitude to the hypotenuse is half the hypotenuse, we know that the hypotenuse is