"Paper Football" is a game that people often play involving a piece of paper folded into a triangle. To score a touchdown, you have to slide your "football" across a desk to your opponent's side. However, the football must be partway past the edge of the table, but not so far past that it falls. This is called the "end zone". If more than half of the football is past the edge of the table, it will fall off and you won't get a touchdown. So a diagram of the farthest the football can be over the edge of the table so that you can score would look like this:

You have decided to play a game of paper football with your friends, and proceed to create your football by repeatedly folding a piece of paper. When you are finished, the football is triangle that has a right angle and two other angles that are the same as each other. You decide to figure out what the maximum distance is that the football can be over the edge without falling over. This is half the length of the hypotenuse. You measure the length of one of the shorter sides of the triangle and find that it is 2 cm long.

Can you figure out what the length of half of the hypotenuse is with just this information?

### Isosceles Right Triangles

An isosceles right triangle is an isosceles triangle and a right triangle. This means that it has two congruent sides and one right angle. Therefore, the two congruent sides must be the legs.

Because the two legs are congruent, we will call them both \begin{align*}a\end{align*} and the hypotenuse \begin{align*}c\end{align*}. Plugging both letters into the Pythagorean Theorem, we get:

\begin{align*}a^2 + a^2 & = c^2\\ 2a^2 & = c^2\\ \sqrt{2a^2} & = \sqrt{c^2}\\ a\sqrt{2} & = c\end{align*}

From this we can conclude that the hypotenuse length is the length of a leg multiplied by \begin{align*}\sqrt{2}\end{align*}. Therefore, we only need one of the three lengths to determine the other two lengths of the sides of an isosceles right triangle. The ratio is usually written \begin{align*}x:x:x\sqrt{2}\end{align*}, where \begin{align*}x\end{align*} is the length of the legs and \begin{align*}x\sqrt{2}\end{align*} is the length of the hypotenuse.

#### Measuring the Sides of Triangles

1. Find the lengths of the other two sides of the isosceles right triangle below.

If a leg has length 8, by the ratio, the other leg is 8 and the hypotenuse is \begin{align*}8\sqrt{2}\end{align*}.

2. Find the lengths of the other two sides of the isosceles right triangle below.

If the hypotenuse has length \begin{align*}7\sqrt{2}\end{align*}, then both legs are 7.

3. Find the lengths of the other two sides of the isosceles right triangle below.

Because the leg is \begin{align*}10\sqrt{2}\end{align*}, then so is the other leg. The hypotenuse will be \begin{align*}10\sqrt{2}\end{align*} multiplied by an additional \begin{align*}\sqrt{2}\end{align*}.

\begin{align*}10\sqrt{2} \cdot \sqrt{2} = 10 \cdot 2 = 20\end{align*}

### Examples

#### Example 1

Earlier, you were asked to figure out what the maximum distance is that the football can be over the edge without falling over.

With your knowledge of the ratios of an isosceles right triangle, you know that the hypotenuse is equal to \begin{align*}\sqrt{2}\end{align*} times the length of each of the other sides. Since it is known that the length of the other side is 2 cm, you therefore know that the length of the hypotenuse is \begin{align*}2\sqrt{2}\end{align*} cm. However, since the football will fall off of the table if it is more than halfway over the edge, the farthest the football can go off of the table is \begin{align*}\frac{2\sqrt{2}}{2} = \sqrt{2} \approx 1.41\end{align*} cm.

#### Example 2

Find the length of the other two sides of the isosceles right triangle below:

Since we know the length of the given leg is \begin{align*}12\end{align*}, and it isn't the hypotenuse, that means the other side that isn't opposite the right angle also has a length of \begin{align*}12\end{align*}. We can then determine from the relationships for an isosceles right triangle that the length of the hypotenuse is \begin{align*}12\sqrt{2}\end{align*}.

#### Example 3

Find the length of the other two sides of the isosceles right triangle below:

Since we know the length of the hypotenuse is \begin{align*}\sqrt{8}\end{align*}, we can determine the lengths of the other two sides. Because the length of the hypotenuse is \begin{align*}\sqrt{2}\end{align*} times the length of the other sides, we can construct the following:

\begin{align*} x \sqrt{2}= \sqrt{8}\\ x = \frac{\sqrt{8}}{\sqrt{2}}\\ x = \sqrt{4} = 2 \end{align*}

#### Example 4

Find the length of the other two sides of the isosceles right triangle below:

** **

Since we know the length of the given leg is \begin{align*}\sqrt{2}\end{align*}, the length of the hypotenuse is then \begin{align*}\sqrt{2} \times \sqrt{2} = \sqrt{4} = 2\end{align*}. The length of the other side is the same as the given side, so it is \begin{align*}\sqrt{2}\end{align*}.

### Review

- In an isosceles right triangle, if a leg is \begin{align*}3\end{align*}, then the hypotenuse is __________.
- In an isosceles right triangle, if a leg is \begin{align*}7\end{align*}, then the hypotenuse is __________.
- In an isosceles right triangle, if a leg is \begin{align*}x\end{align*}, then the hypotenuse is __________.
- In an isosceles right triangle, if the hypotenuse is \begin{align*}16\sqrt{2}\end{align*}, then each leg is __________.
- In an isosceles right triangle, if the hypotenuse is \begin{align*}12\sqrt{2}\end{align*}, then each leg is __________.
- In an isosceles right triangle, if the hypotenuse is \begin{align*}22\end{align*}, then each leg is __________.
- In an isosceles right triangle, if the hypotenuse is \begin{align*}x\end{align*}, then each leg is __________.
- A square has sides of length 16. What is the length of the diagonal?
- A square’s diagonal is \begin{align*}28\sqrt{2}\end{align*}. What is the length of each side?
- A square’s diagonal is 28. What is the length of each side?
- A square has sides of length \begin{align*}3\sqrt{2}\end{align*}. What is the length of the diagonal?
- A square has sides of length \begin{align*}6 \sqrt{2}\end{align*}. What is the length of the diagonal?
- A square has sides of length \begin{align*}4 \sqrt{3}\end{align*}. What is the length of the diagonal?
- A baseball diamond is a square with 80 foot sides. What is the distance from home base to second base? (HINT: It’s the length of the diagonal).
- Four isosceles triangles are formed when both diagonals are drawn in a square. If the length of each side in the square is \begin{align*}s\end{align*}, what are the lengths of the legs of the isosceles triangles?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.5.