Ykhjuy

I was doing a problem in which I was told the lengths of all sides of an isosceles triangle and asked to find the area.

I solved the problem by dividing the isosceles triangle into two equal triangles to find the height which I used in the area formula for the original triangle.

Looking at the answer, my method resulted in the correct value but, it seems I could have used the legs of the isosceles triangle (both 8) as the base and height and skipped finding the height. Why does that shortcut work?
The question is below:

7. 
   

   If the hypotenuse of an isosceles right triangle is $8 sqrt 2$, what is the area of the triangle?

   
   
   
   
   
   • (A) 18
   
   
   • (B) 24
   
   
   • (C) 32
   
   
   • (D) 48
   
   
   • (E) 64
   
   
   
   

Pythagoras' theorem ($a^2 + b^2 = c^2$) is generally used to compute one "missing" side when two others are known in a triangle known to be a right triangle.

However, there is a converse of Pythagoras' theorem, whereby you can show a given triangle is right if the side lengths satisfy the relation.

In this case, $8^2 + 8^2 = (8sqrt 2)^2$, so the triangle is right-angled, and you can immediately find the area as $frac 12 (8)(8) = 32$.

(In this case, you don't need the converse. It's already given to be a right triangle. But in the case you're given the side lengths of $(8,8,8sqrt 2)$, you can apply the converse to simplify your work a bit).

After the edit to the OP, yeah, as pointed out by Deepak in the comments: it is because the triangle is not just any isosceles triangle, but an isosceles right triangle. That the question specifies this also may be indicative that your "shortcut" was the intended method (though kudos to you for finding an additional method either way!).

As is probably obvious whenever you draw right triangles, its area can be given by

$$A = frac12 ab$$

where $a,b$ are the legs of the triangle. In right triangles, the legs can be used as the height and the base. To actually further this discussion and extend to isosceles right triangles, suppose you have only the hypotenuse $h$. Since the triangle is isosceles and right, the legs are equal ($a=b$) and are given by $h/sqrt 2$. In that light we could make this even shorter by noting:

$$A = frac12 ab = frac12 frac{h}{sqrt 2} frac{h}{sqrt 2} = frac{h^2}{4}$$

so in a sense you don't even need to find the legs: in an isosceles right triangle, the hypotenuse uniquely determines the legs, and vice versa.

If you know the lengths $L_1$, $L_2$ of two legs of a triangle and the angle between them $theta$, then you can calculate the area as
$$A = frac12L_1L_2sintheta$$
In your case $theta$ was 90 degrees, making the formula $A = frac12L_1L_2$.