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Textbook
1. Welcome
2. Vocabulary approach
3. Quantitative reasoning
3.1 Quant intro
3.2 Arithmetic & algebra
3.3 Statistics and data interpretation
3.4 Geometry
3.4.1 Angles
3.4.2 Triangle basics
3.4.3 Sum of interior angles
3.4.4 Pythagorean theorem
3.4.5 Right triangles (45-45-90)
3.4.6 Right triangles (30-60-90)
3.4.7 Triangle inequality theorem
3.4.8 Coordinate plane
3.4.9 Equation for a line
3.4.10 Graphing inequalities
3.4.11 Graphing parabolas
3.4.12 Graphing circles
3.4.13 Parallel and perpendicular lines
3.4.14 Quadrilaterals
3.4.15 Circles
3.4.16 3D shapes
3.4.17 Polygons
3.4.18 Regular polygons
3.4.19 Shaded region problems
3.5 Strategies
4. Verbal reasoning
5. Analytical writing
6. Wrapping up
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3.4.6 Right triangles (30-60-90)
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3. Quantitative reasoning
3.4. Geometry

Right triangles (30-60-90)

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Now that you’ve learned about 45-45-90 triangles, it should be no surprise that a triangle with angle measurements of 30, 60, and 90 degrees makes… a 30-60-90 triangle. These triangles look like this:

30 60 90 right triangle

The ratio of the side lengths of a 30-60-90 triangle is always x:x3​:2x, and this is always consistent regardless of the size of the triangle.

The hypotenuse of a 30-60-90 triangle is always double the smallest side length, and the third side is the smallest side length multiplied by 3​.

These values can also be derived from the Pythagorean theorem. Imagine if the x value in this triangle was 1:

a2+b212+3​21+34​=c2=22=4=4​

Sidenote
An equilateral triangle split in half makes two 30-60-90 triangles

This is another shape-splitting trick you’ll want to remember.

This one is a little more complicated even when drawn out, but take a moment to understand the figure below. We’ve made two 30-60-90 triangles by folding this equilateral triangle in half.

30 60 90 right triangle bisected equilateral triangle

Example 30-60-90 triangle question

Now that we know the facts, let’s try an exam question!

The triangle below has an area of 183​.
What is the perimeter of the triangle? Unlabeled special 30-60-90 right triangle A. 18+62​
B. 363​
C. 36
D. 24
E. 18+63​

See if you can solve it yourself, then check your answer!

(spoiler)

Answer: E. 18+63​

The figure gives us enough information to guarantee that the shape is a 30-60-90 triangle. For every triangle, the sum of the interior angles is 180∘, and since we’re given the right angle of 90∘ and another angle of 60∘, the last unknown angle must be 30∘.

The sides of any 30-60-90 triangle have a ratio of x:x3​:2x, and the area of a triangle is A=bh/2. Matching this up with our 30-60-90 triangle, our base b=x, and the height h=x3​.

We can plug these variables into the area equation and solve for the length of the side x.

Area183​363​36366​=bh/2=x(x3​)/2=x(x3​)=x(x)=x2=x​

Now that we know the short side x=6, we can apply the ratio of x:x3​:2x to get all three sides of 6:63​:12.

The final step is to calculate the perimeter by adding up the sides.

P​=6+63​+12=18+63​​

Memorize these special triangle ratios and you’ll be able to solve triangle questions with ease!

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