# How do you find the area of a isosceles triangle with base 10 and perimeter 36?

May 2, 2018

The area is $60$ square units

#### Explanation:

Giving us the perimeter of an isosceles triangle means we are indirectly given the lengths of the three sides.

If the base is $10$, the the other two equal sides add to $26$:

$36 - 10 = 26$

The equal sides are therefore $26 \div 2 = 13$

To find the area of the triangle we need its height.

the height of an isosceles triangle can be found from its line of symmetry. If you cut the triangle into two identical triangles, for each we will have:

A 90° angle

A base of $5 \text{ } \rightarrow \left(10 \div 2\right)$

The hypotenuse is $13$

You can therefore use Pythagoras' Theorem to find the third side which will be the height of the triangle.

You might recognise that these are two values of the Pythagorean Triple, $5 , 12 , 13 \text{ } \leftarrow$ the height is $12$.

If not you can work it out.

${x}^{2} = {13}^{2} - {5}^{2}$
${x}^{2} = 144$
$x = \sqrt{144}$
$x = 12$

Now we know that the height is $12$ so we can find the area of the original triangle.

$A = \frac{b h}{2}$

$A = \frac{10 \times 12}{2}$

$A = 60$ square units

May 2, 2018

$a = 60$

#### Explanation:

Let;

$a \to \text{area}$

$b \to \text{base}$

$p \to \text{perimeter}$

$s \to \text{slant height}$

$h \to \text{height}$

$p = 36$

$b = 10$

Recall;

$a = \frac{1}{2} b \times h$

$p = b + 2 s$

Firstly;

$p = b + 2 s$

Substituting the values..

$36 = 10 + 2 s$

$36 - 10 = 2 s$

#26 = 2s

$\frac{26}{2} = s$

$13 = s$

Therefore, slant height is $13$

Also recall;

Using Pythagoras theorem;

Since we are using pythagoras theorem, the base of the triangle would be halved..

Therefore, $b = \frac{10}{2} = 5$

${s}^{2} = {b}^{2} + {h}^{2}$

Substituting the values..

${13}^{2} = {5}^{2} + {h}^{2}$

$169 = 25 + {h}^{2}$

$169 - 25 = {h}^{2}$

$144 = {h}^{2}$

$\sqrt{144} = h$

$12 = h$

Hence;

$a = \frac{1}{2} b \times h$

Substituting the values..

$a = \frac{1}{2} \times 10 \times 12$

$a = 5 \times 12$

$a = 60$