# Thread: Deriving hypotenuse from perimeter of a triangle

1. ## Deriving hypotenuse from perimeter of a triangle

The perimeter of a certain isosceles right triangle is 16+16√2. What is the hypotenuse of the triangle?

2. Right triangle with side lengths a, b, c (where c is hypotnuse)
Isosceles means a=b

perimeter = a + b + c = a + a + c
Pythagorean theorem says: a^2 + b^2 = a^2 + a^2 = c^2

You have 2 equations and 2 unknowns. Solve for c.

3. first draw the triangle. If the hypotenuse is "b" and the 2 other equal sides are "a" then the perimeter is:

$\displaystyle b+2a=16+16\sqrt{2}$

You know the angles of an isosceles right triangle, there are 2 45 degree angles $\displaystyle \frac{\pi}{4}$ and one angle $\displaystyle \frac{\pi}{2}$

So the sine of pi over 4 is $\displaystyle sin(\frac{\pi}{4})=\frac{a}{b}$ solve this for "a" and plug that into the original equation for the perimeter and use algebra to solve for "b".

4. Snowtea's solution is easier, I didn't catch that...

5. Hello, mtylerrose!

$\displaystyle \text{The perimeter of a certain isosceles right triangle is }16+16\sqrt{2}$
$\displaystyle \text{What is the hypotenuse of the triangle?}$

You should know that the sides of an isosceles right triangle are: .$\displaystyle x,\:x,\:x\sqrt{2}$

The perimeter is: .$\displaystyle x + x + x\sqrt{2} \;=\;16 + 16\sqrt{2}$

. . . . . . . . . . . . . . . $\displaystyle (2 + \sqrt{2})x \:=\:16(1+\sqrt{2})$

. . . . . . . . . . . . . . . . . . . . . $\displaystyle x \;=\;\dfrac{16(1+\sqrt{2})}{2+\sqrt{2}}$

Rationalize: .$\displaystyle \displaystyle x\;=\;\frac{16(1+\sqrt{2})}{2 + \sqrt{2}} \cdot\frac{2-\sqrt{2}}{2-\sqrt{2}} \;=\;\frac{16(2 - \sqrt{2} + 2\sqrt{2} - 2)}{4 - 2}$

. . . . . . . . . $\displaystyle x \;=\;\dfrac{16\sqrt{2}}{2} \;=\;8\sqrt{2}$

The hypotenuse is: .$\displaystyle x\sqrt{2} \;=\;8\sqrt{2}\cdot\sqrt{2} \;=\;16$

6. Thanks everyone! I had no idea where to start, but now I am starting to understand.

Soroban,

How did you know to rationalize the way you did?

7. Originally Posted by mtylerrose
Thanks everyone! I had no idea where to start, but now I am starting to understand.

Soroban,

How did you know to rationalize the way you did?
That's a general technique he used that's taught in algebra courses. You always multiply by the conjugate of the denominator to rationalize an expression like that.