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Today, we will be covering special right triangles, their rules and some problems you might encounter that deal with these.

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So what are special right triangles?

The special right triangles are right triangles with angle measures of 30-60-90 and 45-45-90. We call these two triangles the 30-60-90, and the 45-45-90 special right triangles.

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So what makes these triangles special? Well, we can express the side lengths as a ratio of each other.

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For example, if I know that the hypotenuse of this triangle is 14, because its a 30-60-90 triangle, then I also know that the side length across from the 30 degree angle is equal to 7, and the side length across from the 60 degree angle is 7 times the square root of 3.

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We know this because we can use a ratio to describe the side lengths of these special right triangles.

The ratio for a 30-60-90 triangle tells us that the side across from the 30 degree angle is equal to *a*, the side across from the 60 degree angle is *a *times the square root of 3 and the hypotenuse is equal to 2*a*.

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In this case, we know that the hypotenuse is 14. This means we can say that 14 = 2* a*.

We can solve for a by dividing both sides of the equation by 2. This tells us that

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The 45-45-90 right triangle is also called the isosceles right triangle. Here is the ratio of side lengths for the isosceles right triangle.

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Both the sides across from a 45 degree angle are both going to be equal; we call both these sides *a*. Our ratio tells us that the sides across from the 45 degree angles are equal to *a*, and the hypotenuse, or the side across from the 90 degree angle is going to be equal to *a* times the square root of 2.

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Now that we have these ratios in mind, let’s take a look at some practice problems.

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Let’s take a look at this problem. Here, we are given a right angle triangle with a 60 degree angle. Since the sum of angles in a triangle must be 180 degrees, we know that the third angle must be equal to 30 degrees. This means that we are dealing with a 30-60-90 right triangle. To solve for *h*, we should use the ratio of side lengths.

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Since we are given the side across from the 60 degree angle, we should set that side length equal to *a* times the square root of 3.

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We can solve for *a* by dividing both sides by the equation by the square root of 3. This tells us that *a* is equal to 17.

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Now that we know the value of *a*, we can plug it into our ratio to find side length *h*.

Since side *h* is the hypotenuse, we are looking for the value of 2 times *a*.

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Two times seventeen is equal to thirty-four.

This means that the hypotenuse of this triangle is 34.

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Let’s take a look at a slightly more complex problem.

In this case, we are looking for the length of side *h* and we are given two angles and a side length.

We also know that the green triangle has two sides that are equal.

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The triangle on the left has a 30 degree angle and a 90 degree angle. Since the sum of interior angles of a triangle is 180 degrees, we know that the third angle in this triangle must be 60 degrees. This means that the triangle on the left is a 30-60-90 special right triangle.

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We can use the ratio of side lengths of a 30-60-90 triangle to find the other side lengths of the blue triangle.

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Since the side length that has a value of 36 is opposite the 90 degree angle, this is our hypotenuse and we can set it equal to 2 times *a*.

If 2*a* is equal to 36, then *a* is equal to 18.

This means that the side across from our 30 degree angle has a value of 18.

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Because of this, we know that both the congruent sides of the green triangle will also have a length of 18.

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As we should know, the sum of angles along a straight line add up to 180 degrees. Since we know one of our angles is 90 degrees, the angle along the same straight line will also be 180 degrees. This means that our green triangle is also a right triangle.

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Since it has two congruent sides, it is our isosceles-right triangle, or the 45-45-90 right triangle.

We already know two of the sidelengths for our green triangle.

We know that both of our congruent sides will be 18.

Because of this, we know *a*.

We can solve for our hypotenuse by plugging in *a* to find our hypotenuse.

In this case, the value of *h* is 18 times the square root of 2.

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Let’s take a look at one more problem you may encounter.

Here, we want to find the area of an equilateral triangle, but we are only given one of its side lengths.

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Since we are dealing with an equilateral triangle, we know that all of our side lengths will be equal.

We also know that equilateral triangles have three 60 degree angles since the sum of angles add up to 180 degrees.

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To find the area of any triangle, we use the formula: the area is equal to one-half times the base times the height.

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We already know that the base of this triangle is equal to 8, so we only need to find the height in order to be able to solve for the area.

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The height of an equilateral triangle cuts it exactly in half, and meets the base at a 90 degree angle.

It also cuts the central angle of the triangle in half.

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This means that we form a 30-60-90 right triangle as we try to find the height of an equilateral triangle.

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The height of our equilateral triangle is the side length opposite the 60 degree angle of our 30-60-90 triangle.

We can use our ratio for the 30-60-90 triangle to find the height of our equilateral triangle.

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Since the side length that is equal to 4 is opposite the 30 degree angle, we know that *a* = 4.

This means that the height of our equilateral triangle is 4 times the square root of 3.

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Now that we have our base, and now that we have our height, we can plug these values in to solve for our area.

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The area is one half times eight times four times the square root of three, which is equal to sixteen times the square root of three.

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To recap, special right triangles have unique ratios that allow us to find all of the sidelengths as long as we know one of them.

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I hope this video has been helpful, and be sure to check out our other content. Thank you!

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