If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## High school geometry

### Course: High school geometry>Unit 5

Lesson 3: Special right triangles

# 30-60-90 triangle example problem

Using what we know about 30-60-90 triangles to solve what at first seems to be a challenging problem. Created by Sal Khan.

## Want to join the conversation?

• In the previous 30-60-90 video I thought I understood the ratio of the angles to be x: square root of 3 over 2 times x : and x over 2. At Sal says the ratio is 1: square root of 3 : 2. Can someone explain this please? • They're the same thing, it's just that the former is expressed in terms of x (which is the hypotenuse). But first, you have to be careful: since we're only dealing with ratios, you can order the three sides as you please, but it's better to order from shortest to longest side, otherwise it might get confusing. So here, it would be better to say x/2 : (sqrt(3)/2)x : x. This is consistent with 1 : sqrt(3) : 2, shortest to longest. Also, you might or might not be a bit confused by the notation I used, but I'm sure if you compare it with what you know about the ratios, you'll be fine. But I'd suggest you to be careful with how you write mathematical expressions; you wrote "square root of 3 over 2 times x", but this is actually very ambiguous and can be interpreted in at least two ways, one of which means something pretty different from what you want!

Anyway, as I said, they're the same. If you take a closer look at 1 : sqrt(3) : 2, you'll notice that the length of the longest side, which is 2, is two times that of the shortest side, which is 1. So if you say the longest side is x, then the shortest side is x/2. Also, we know that the ratio of the shortest side to the middle one is 1 : sqrt(3). In other words, you have to multiply the shortest side by sqrt(3) in order to get the middle side. So:

sqrt(3) * x/2 = (sqrt(3)/2)x. You could also write is sqrt(3)*x/2.

But anyway, yeah, 1 : sqrt(3) : 2, and x/2 : (sqrt(3)/2)x2 : x are the exact same thing!
• Is it not true that the middle Δ BDE is an Isoceles triangle? Since Angle EBD= 30 deg.
Then why not use the 2/√3 is the measure of ED. Adding 2/√3 +2/√3 +2 = 4/√3 + 2 = perimeter of triangle BDE. • At , Sal says that the 60-degree side is going to be √3 times 1, but I don't understand how he got √3. • I still don't get how Sal got a square root of 3/3 from 1/ square root of 3. • I don't understand how Sal got a side length measured to the square root of 3 at Anyone? • At the very end, the perimeter was 1/sqrt3 + sqrt3 + 2, then you multiplied by sqrt3/sqrt3 (1) to make 1/sqrt3 into sqrt3 / 3. So with the rest of the numbers, how did sqrt3 become 3 times sqrt3 / 3, what did you multiply the sqrt 3 by? • At , how can we know that side DC is equal to 1 if we only know that side AB is equal to 1 in quadrilateral ADCB? • We know that the quadrilateral has 4 right angles, meaning that it must be a rectangle or a square. Squares and rectangles both have the property that two opposite sides must have the same length (But squares have the additional element of all sides having the same length).

In this problem, we already established that it is either a square or a rectangle. Since we know side AB has a length of 1, and that side DC is opposite of side AB, we also know that side DC has a length of 1. We also know that side AD will have the same length as side BC, using the same property of rectangles.
• I am having trouble understanding how to solve a problem if you're given that the hypotenuse is x and the triangle is a 30-60-90 triangle and the side adjacent to the 30 degree angle is 15 units long. How can I find the hypotenuse? Is there a video on this? • The ratio of the side lengths of a 30-60-90 triangle is 1 ∶ √3 ∶ 2

This means that if the shortest side, i.e., the side adjacent to the 60° angle, is of length 𝑎,
then the length of the side adjacent to the 30° angle is 𝑎√3,
and the length of the hypotenuse is 2𝑎

In this case we have 𝑎√3 = 15 ⇒ 𝑎 = 5√3
Thereby the length of the hypotenuse is 2 ∙ 5√3 = 10√3 ≈ 17.3 units  