When the Pythagorean theorem fails, special right triangles are your new best friend.

Jenn, Founder Calcworkshop^{®}, 15+ Years Experience (Licensed & Certified Teacher)

*It’s true!*

That’s what today’s geometry lesson is all about, so let’s get started.

## What Are Special Right Triangles

Two very special right triangle relationships will continually appear throughout the study of mathematics:

**45-45-90**Triangle**30-60-90**Triangle

In an isosceles right triangle, the angle measures are 45°-45°-90°, and the side lengths create a ratio where the measure of the hypotenuse is sqrt(2) times the measure of each leg as seen in the diagram below.

45-45-90 Triangle Ratio

And with a 30°-60°-90°, the measure of the hypotenuse is two times that of the leg opposite the 30° angle, and the measure of the other leg is sqrt(3) times that of the leg opposite the 30° also seen in the diagram below.

30-60-90 Triangle Ratio

Together we will look at how easy it is to use these ratios to find missing side lengths, no matter if we are given a leg or hypotenuse. Moreover, we will discover that no matter the size of our special right triangle, these ratios will always work.

But why do we need them if we have the Pythagorean theorem for finding side lengths of a right triangle?

Well, one of the greatest assets to knowing the special right triangle ratios is that it provides us with an alternative to our calculations when finding missing side lengths of a right triangle.

Rather than always having to rely on the Pythagorean theorem, we can use a particular ratio and save time with our calculations as Online Math Learning nicely states.

Additionally, there are times when we are only given one side length, and we are asked to find the other two sides.

The Pythagorean theorem requires us to know two-side lengths; therefore, we can’t always rely on it to solve a right triangle for missing sides. Consequently, knowing these ratios will help us to arrive at our answer quickly, but will also be vital in many circ*mstances.

## How To Solve Special Right Triangles

### Example #1

Solve the right triangle for the missing side length and hypotenuse, using 45-45-90 special right triangle ratios.

Solving a 45 45 90 Triangle for Side Lengths

### Example #2

Solve the right triangle for the missing side lengths, using special right triangle ratios.

Special Right Triangles with Radicals

In the video below, you will also explore the 30-60-90 triangle ratios and use them to solve triangles. Additionally, you will discover why it’s very important on how you choose your side lengths. (*HINT: Order Matters!*)

## Common Questions

**Q**: How to find the hypotenuse in special right triangles?**A**: The hypotenuse is always the longest side of a right triangle. We can find the hypotenuse by using the Pythagorean theorem or trigonometric ratios by fist ordering side lengths in increasing value, as seen in the video.

**Q**: How to do multi-step special right triangles?**A**: If we are given a right triangle with one acute angle and side length known, we will first utilize our special right triangle ratios to find one missing side length (either a leg or hypotenuse). Then we will use the Pythagorean theorem to find the remaining side length.

**Q**: What is the 3:4:5 triangle rule?**A**: The 3-4-5 triangle rule uses this well known pythagorean triple. In other words, 3:4:5 refers to a right triangle with side length of 3, 4, and 5, where the hypotenuse is the length of 5 and the legs are 3 and 4, respectively. Consequently, if we are given these three side lengths we know it refers to a right triangle. Additionally, all multiples are also right triangles. For example, 30:40:50 or 6:8:10 are both multiples of 3:4:5 and both indicate right triangle measurements.

**Q**: How do you know if it’s a pythagorean triple?**A**: A right triangle whose side lengths are all positive integers, such as a 3:4:5 triangle or 5:12:13 triangle or 7:24:25 triangle.

**Q**: How to use pythagorean theorem with only one side?**A**: If only one side length is known, we are unable to use the Pythagorean theorem. Therefore, we must first use our trigonometric ratios to find a second side length and then we can use the Pythagorean theorem to find our final missing side.

## Video – Lesson & Examples

1 hr 6 min

- Introduction
**00:00:22**– Overview of the 45-45-90 and 30-60-90 Triangles**Exclusive Content for Member’s Only**

**00:10:39**– Given the special right triangle, find the unknown measures (Examples #1-6)**00:22:20**– Find the missing measures for the given problems (Examples #7-11)**00:36:27**– How to find trig ratios (Examples #12-15)**00:49:42**– Find the indicated measure given an equilateral triangle and square (Examples #16-17)**00:57:50**– Solve the word problem (Examples #18-19)**Practice Problems**with Step-by-Step Solutions**Chapter Tests**with Video Solutions

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