45-45-90直角三角形：边之比

A special type of right triangle is the $45$-$45$-$90$ right triangle. As you can tell from the name of the triangle, this right triangle has interior angles of $45^{\circ}$, $45^{\circ}$, and $90^{\circ}$.

Since two of the angles of this triangle are equal to $45^{\circ}$, the triangle is an isosceles triangle. Being a right triangle, this means that its two legs (the sides of the right angle) have equal length.

Suppose the length of each leg of the $45$-$45$-$90$ triangle is $a$ units.

By the Pythagorean Theorem, we can derive the length of the hypotenuse, denoted as $b$: $$\begin{align} b^2 & = a^2 + a^2 \qquad\\ & = 2a^2\\ \therefore b & = \sqrt{2}a\ \text{units} \end{align}$$

Therefore, the ratios of the sides of a $45$-$45$-$90$ right triangle can be expressed $a:a:b = 1:1:\sqrt{2}$.

If a triangle is a $45$-$45$-$90$ right triangle, the ratio of the sides (leg:leg:hypotenuse) is $1:1:\sqrt{2}$.

(Video): Isosceles Right Triangles

by Art of Problem Solving

In the video below, Richard Rusczyk shows you how to find the sides of a $45$-$45$-$90$ triangle. Remember, if a triangle has two angles of equal measure, that means that it will also have two sides of equal length.

If you say that one of the legs (the sides that are of equal length) of an isosceles right triangle is length $x$, we'll always know that the other leg is of length $x$ and the hypotenuse is equal to $x\sqrt{2}$. In other words, the ratios of the sides will be:

$$\text{Leg 1 : Leg 2 : Hypotenuse}\\ x \,\,\,\,\,\,\, : \,\,\,\,\,\,\, x \,\,\,\,\,\,\, : \,\,\,\,\,\,\, x\sqrt{2}$$

All isosceles right triangles are similar to each other, so that ratio will work for every single isosceles right triangle's sides.

To find the area of of a triangle, we need to multiply its base times its height (and then divide by $2$). In an isosceles right triangle, the base and height are the legs!

We know that if the length of one of the legs is $x$, then the hypotenuse's length is $x\sqrt{2}$. But we know the length of the hypotenuse, so let's try to find an equation that tells us the length of a leg in terms of the hypotenuse (which we'll call $h$):

$$h = x\sqrt{2}\\ \frac{h}{\sqrt{2}} = x$$

So let's apply this to the triangle we've got. Each leg will measure:

$$\frac{25}{\sqrt{2}} = x$$

Now we can just square that number, divide by $2$, and we'll be done!

$$\text{area} = (\frac{25}{\sqrt{2}})^2\div 2\\\ \\ \text{area} = (\frac{625}{2})\div 2\\\ \\ \text{area} = (312.5) \div 2\\\ \\ \text{area} = 156.25$$