An isoceles right triangle has two congruent legs.  We use the Pythagorean Theorem, which states that a² + b² = c², where a and b are the legs and c is the hypotenuse.

Let $\displaystyle x$ = leg length.

Because this is an isosceles triangle, the two legs have the same length. Plug this and the hypotenuse length into the Pythagorean Theorem and solve for x:

$\displaystyle x^{2}+x^{2}=6^{2}$ 

$\displaystyle 2x^{2}=36$ 

$\displaystyle x = 3\sqrt{2}\; cm$

Thus the perimeter is $\displaystyle P=2x +6$.

Plug in our value for x: $\displaystyle P=6\sqrt{2} + 6\; cm$

The perimeter of a triangle is the distance around the outside, and can be found by adding the side lengths together: $\displaystyle a+b+c$.

The first point of business is to realize is that we have a 45-45-90 triangle. That means the two legs of our triangle are congruent and are thus each 8. We can find our hypotenuse by multiplying by $\displaystyle \sqrt{2}$. Thus our hypotenuse is $\displaystyle 8\sqrt{2}$.

Our perimeter is simply the sum of of the lengths of our three sides.

$\displaystyle 8+8+8\sqrt{2}=16+8\sqrt{2}$

We are given that $\displaystyle \Delta ABC$ is a $\displaystyle 45^\circ-45^\circ-90^\circ$ triangle and that $\displaystyle \overline{AC}=8$. Calculate the length of sides $\displaystyle \overline{AB}$ and $\displaystyle \overline{BC}$ using either the Pythagorean Theorem or the $\displaystyle 1:1:\sqrt{2}$ ratio of the sides of a $\displaystyle 45^\circ-45^\circ-90^\circ$ triangle.

Using the Pythagorean Theorem:

Because $\displaystyle \Delta ABC$ is a $\displaystyle 45^\circ-45^\circ-90^\circ$ triangle, we know $\displaystyle \overline{AB}=\overline{BC}$ so we can represent both legs of the triangle with one variable.  Let's use $\displaystyle a$.

$\displaystyle a^2+a^2=8^2$

$\displaystyle 2a^2=64$

$\displaystyle a^2=32$

$\displaystyle a=\sqrt{32}=\sqrt{16\cdot 2}=4\sqrt{2}$

Using the ratio of the sides:

Divide the length of the hypotenuse by $\displaystyle \sqrt{2}$.

$\displaystyle \frac{8}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}}=\frac{8\sqrt{2}}{2}=4\sqrt{2}$

Either way, the length of $\displaystyle \overline{AB}=\overline{BC}=4\sqrt{2}$

Calculate the perimeter by adding the length of all the sides.

$\displaystyle \overline{AB}+\overline{BC}+\overline{AC}=perimeter$

$\displaystyle 4\sqrt{2}+4\sqrt{2}+8=8\sqrt{2}+8$

This can be factored to $\displaystyle 8(\sqrt{2}+1)$.

The problem tells us the triangle is 45/45/90. The goal is to solve for the perimeter, which can be determined through $\displaystyle s_{1}+s_{2}+s_3=P$, where the s's are in reference to the three sides and P stands for perimeter. 

In the figure, two of the three sides are given. In order to calculate the hypotenuse, two methods are possible:

1. using the Pythagorean Theorem 

2. using 

After calculations, the hypotenuse is $\displaystyle 12\sqrt{2} ft$

Perimeter can be calculated out to be:

$\displaystyle P= 12+12+12\sqrt{2}$

$\displaystyle P=24+12\sqrt{2}$

$\displaystyle P={\color{Orchid} 40.97}ft$

In order to solve for the perimter (the sum of all sides), all side lengths must be known. 

Because it's been stated the triangle is 45/45/90, this means that it is also isosceles. Therefore, given that one of the leg lengths is 5 cm, this means that the other leg must also be 5 cm. This leaves the hypotenuse as unknown; let's label this as x. 

The third side can be easily determined through the Pythagorean Theorem because it's a right triangle. 

$\displaystyle a^2+b^2=c^2$, c=x

$\displaystyle 5^2+5^2=x^2$

$\displaystyle 25+25=x^2$

$\displaystyle 50=x^2$

$\displaystyle \sqrt{50}=\sqrt{x^2}$

$\displaystyle \pm 5\sqrt{2}=x$

But because the hypotenuse measures distance, x cannot be a negative number. Therefore, x=5√2.

Now, perimeter can be solved for.

$\displaystyle 5+5+5\sqrt{2}=P$

$\displaystyle {\color{Blue} 10+5\sqrt{2}=P}$

To find the perimeter we must find the hypotenuse and then sum all side lengths to find the perimeter. Remember the hypostenuse of an isoceles right triangle is the side length multiplied by the square root of $\displaystyle 2$.

Thus,

$\displaystyle P=s+s+s\sqrt{2}=5+5+5\sqrt{2}=10+5\sqrt2$

Recall how to find the perimeter of a triangle:

$\displaystyle \text{Perimeter}=a+b+c$

Now, because this is a right isosceles triangle, we know the following:

$\displaystyle a=b$

From the information given in the question, we already have two of the sides needed. Now, recall the Pythagorean Theorem to find the third side of the triangle, the hypotenuse.

$\displaystyle a^2+b^2=c^2$

The Pythagorean Theorem can then be simplifed to the following equation:

$\displaystyle a^2+a^2=2a^2=c^2$

Now, solve for $\displaystyle c$ since the question asks for the length of the hypotenuse.

$\displaystyle c=\sqrt{2a^2}$

$\displaystyle c=a\sqrt2$

Now, plug in the given value for $\displaystyle a$ to find the length of the hypotenuse.

$\displaystyle c=12\sqrt2$

Now that we have all three sides of the triangle, we can find the perimeter. Use a calculator and round to $\displaystyle 2$ decimal places.

$\displaystyle \text{Perimeter}=12+12+12\sqrt2=40.97$

Recall how to find the perimeter of a triangle:

$\displaystyle \text{Perimeter}=a+b+c$

Now, because this is a right isosceles triangle, we know the following:

$\displaystyle a=b$

From the information given in the question, we already have two of the sides needed. Now, recall the Pythagorean Theorem to find the third side of the triangle, the hypotenuse.

$\displaystyle a^2+b^2=c^2$

The Pythagorean Theorem can then be simplifed to the following equation:

$\displaystyle a^2+a^2=2a^2=c^2$

Now, solve for $\displaystyle c$ since the question asks for the length of the hypotenuse.

$\displaystyle c=\sqrt{2a^2}$

$\displaystyle c=a\sqrt2$

Now, plug in the given value for $\displaystyle a$ to find the length of the hypotenuse.

$\displaystyle c=5\sqrt2$

Now that we have all three sides of the triangle, we can find the perimeter. Use a calculator and round to $\displaystyle 2$ decimal places.

$\displaystyle \text{Perimeter}=5+5+5\sqrt2=17.07$

Recall how to find the perimeter of a triangle:

$\displaystyle \text{Perimeter}=a+b+c$

Now, because this is a right isosceles triangle, we know the following:

$\displaystyle a=b$

From the information given in the question, we already have two of the sides needed. Now, recall the Pythagorean Theorem to find the third side of the triangle, the hypotenuse.

$\displaystyle a^2+b^2=c^2$

The Pythagorean Theorem can then be simplifed to the following equation:

$\displaystyle a^2+a^2=2a^2=c^2$

Now, solve for $\displaystyle c$ since the question asks for the length of the hypotenuse.

$\displaystyle c=\sqrt{2a^2}$

$\displaystyle c=a\sqrt2$

Now, plug in the given value for $\displaystyle a$ to find the length of the hypotenuse.

$\displaystyle c=4\sqrt2$

Now that we have all three sides of the triangle, we can find the perimeter. Use a calculator and round to $\displaystyle 2$ decimal places.

$\displaystyle \text{Perimeter}=4+4+4\sqrt2=13.66$