What is a 45-45-90 Triangle?
A 45-45-90 triangle can be formed by cutting a square in half. This is one of two special types of right triangles (the other is a 30-60-90 triangle, which is half of an equilateral triangle).
What do you know about a square? All 4 sides are the same length. Since a 45-45-90 triangle is half of a square, this means that the two sides that form the 90 degree angle are the same length.
For this reason, a 45-45-90 triangle is often referred to as an isosceles right triangle. If you know one of the legs of a 45-45-90 triangle, the other leg will be the same.
We know that one leg of this 45-45-90 triangle is 7 units long. The two legs that form the right angle in a 45-45-90 triangle are always the same length. This means that x = 7.
Hypotenuse of a 45-45-90 Triangle
The hypotenuse of a 45-45-90 triangle is not quite as easy to find. There's a long way and a short way to find it. The long way is to use the Pythagorean Theorem. We'll briefly look at the long way first and then show you how to use the shortcut.
The Long Way: Use the Pythagorean Theorem
If you know both the legs of a right triangle, you can use the Pythagorean Theorem to find the hypotenuse. Remember, a and b are the legs that form the right triangle and c is the hypotenuse.
We could leave the answer for the hypotenuse as the square root of 98 or round it and get a decimal answer. To show the shortcut relationship, we can simplify the square root of 98 instead. When you're simplifying a square root, look for factors of the number under the square root that are perfect squares. See if you can divide the number by 4, 9, 16, 25, etc. . In this case, 49 is a perfect square that is a factor of 98. This means we can rewrite the problem as the square root of 49 times the square root of 2. The square root of 49 is 7, so this simplifies to 7 times the square root of 2.
Now look back at the diagram. Do you see the shortcut? How is the length of the hypotenuse related to the length of the two legs?
Hopefully you can see that the hypotenuse is simply the length of one of the legs times the square root of 2. This happens every single time with a 45-45-90 triangle. So from now on, don't use the Pythagorean Theorem to find the hypotenuse. Use the shortcut and just multiply the leg by the square root of 2.
The Shortcut for 45-45-90 Triangles
If you know the leg of a 45-45-90 triangle, you can find the hypotenuse by multiplying the leg by the square root of 2.
Find b and c.
The two legs of a 45-45-90 triangle are always the same length. The legs are the two sides that form the right angle - the 5 and the b. This means that b is also 5. To find the hypotenuse, we can use the shortcut and simply multiply the leg by the square root of 2.
Find d and e.
This one is slightly trickier because of the way the triangle has been drawn. Remember,the legs of a right triangle form the right angle, the form the letter "L". In this triangle, the two legs are d and 8 - they form the 90 degree angle at the top. The legs of a 45-45-90 triangle are always the same length so d = 8. The hypotenuse is always across from the right angle. In this triangle, the hypotenuse is at the bottom of the triangle, the e. To find the hypotenuse of a 45-45-90 triangle, you need to multiply the leg by the square root of 2.
Finding the Legs of a 45-45-90 Triangle
In the earlier examples, we were given one of the legs of a 45-45-90 triangle. What if you're given the hypotenuse instead and you have to find the legs? No problem, you just need to go backwards.
When you know the leg of a 45-45-90 triangle, you multiply it by the square root of 2. How do you undo multiplication? You divide. If you know the hypotenuse, you can just divide it by the square root of 2 to find the leg. Once you have one leg figured out, the second leg will be the same.
Find f and g.
Don't get thrown off by the way the picture has been drawn. The legs of a right triangle always form the letter "L" - they make the 90 degree corner. In this triangle, the legs are f and g. The hypotenuse is always across from the right angle - the 10 is the hypotenuse.
In a 45-45-90 triangle, you can divide the hypotenuse by the square root of 2 to find the leg. Once you know one leg, you know the other because both legs always have the same length.
Your teacher may have you just leave the answer this way. Technically, the answer is not simplified because it has a radical in the denominator of the fraction. If you're asked to simplify the answer, you can multiply the numerator and denominator by the square root of 2. When you do this, you end up with a perfect square under the radical that can be simplified.
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