Question

We have a right angle triangle with length of hypotinos being 15 and the perimeter being 36. The two short sides are not equal length. How do I figure out the formula.

Answer 1

We have a right angle triangle with length of hypotinos being 15 and the perimeter being 36. The two short sides are not equal length. How do I figure out the formula.

Pythagorean theorem:

$$a ^ {2} + b ^ {2} = c ^ {2}$$

where $c$ represents the length of the hypotenuse, and $a$ and $b$ represent the lengths of the other two sides.

In our case $c = 15$, therefore:

$$a ^ {2} + b ^ {2} = 1 5 ^ {2}$$

Perimeter by definition equals:

$$P = a + b + c$$

In our case, $P = 36$, $c = 15$

$$a + b + 15 = 36 \Rightarrow a + b = 21$$

And two short sides are not of equal length: $a \neq b$

So, we have system of equations:

$$\left\{ \begin{array}{c} a ^ {2} + b ^ {2} = 1 5 ^ {2} \\ a + b = 2 1 \end{array} \right.$$

From second: $(a + b)^2 = a^2 + 2ab + b^2 = 21^2$

Therefore: $ab = \frac{21^2 - 15^2}{2} = 108$

$$\left\{ \begin{array}{l} a b = 1 0 8 \\ a + b = 2 1 \end{array} \right.$$

From first: $a = \frac{108}{b}$

Substitute to the second:

$$\frac {1 0 8}{b} + b = 2 1$$

Or:

$$b ^ {2} - 2 1 b + 1 0 8 = 0$$

Solving this equation:

$$b = 9, 1 2 \Rightarrow a = 1 2, 9$$

Therefore, short sides equal 9 and 12.