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The Math section of the SAT is full of special right triangles.
The 5-12-13 triangle is a common triple. If you can recognize the triple pattern then you can easily calculate a missing side. Here is a list of the common triples associated with a 5 -12-13 right triangle.

                   x2  10-24-26
                   x3  15-36-39
                   x4  20-48-52
                   x5 25-60-65

For example:  If you see a right triangle with a leg of 15 and the hypotenuse is 39, you can divide by 3 to get back to the 5, 12, 13 triangle.  Then you take the leg of 12 and multiply it by 3 to find the missing length, 36, of the larger right triangle.  

Common Core Standard F.TF.3
video tutorial special right triangle

What are special right triangles?

white box/special right triangles
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3 4 5  triangle

Special Right Triangles

What are special right triangles? They are triangles that have regular patterns which makes calculations of the triangle easier. The following  special right triangles are investigated, 
  • 30-60-90 triangle 
  •  45-45-90 triangle
  • 3-4-5 triangle
  •  5-12-13 triangles. 

45°- 45°- 90° Triangles


A right triangle with two sides of equal lengths is a 45°- 45°- 90° triangle.


The length of the sides are in the ratio of

1:1: √2


Leg length = 1/2 hypotenuse√2


Hypotenuse = leg√2


30°- 60°- 90° Triangles


Hypotenuse is always opposite the right angle


Short Leg is opposite the 30◦ angle


Long leg is opposite the 60◦ angle


The lengths of the sides of a 30°- 60°- 90° triangle are in the ratio of 1: √3 :2


Short leg = ½ hypotenuse


Long leg = 2* short leg


Hypotenuse = √3 * short side



5-12-13 Triangles


A 5-12-13 triangle is a right-angled triangle whose lengths are in the ratio of 5:12:13.


Notice that 5:12:13 satisfies the Pythagorean theorem and is a common triplet. This can be used to identify leg lengths




3-4-5 Triangles


3-4-5 triangles have leg lengths in the ratio of 3:4:5


If the length of the right triangle are in the ratio 3:4:5 then the other leg lengths can be found easily




45 45 90 triangle
30 60 90 triangle
right triangle

If you have the following information:

A right angle

Length of opposite leg

Length of the hypotenuse

Use Sin





A right angle

Length of adjacent leg

Length of the hypotenuse

Use Cosine





A right angle

Length of opposite leg

Length of the adjacent

Use Tangent






5 12 13 triangle

Trig ratios are used to find missing side lengths and angles of special right triangles. See a summary of trig functions below.

Pythagorean theorem and special right triangles

The Pythagorean Theorem can be used to prove that a 5-12-13 and 3-4-5 are right triangles.
The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the two sides squared.

3^2 + 4^2 = 5^2
9 + 16 =25
Lets try a 5-12-13 triangle
5^2 + 12^2 = 13^2
25 +144 =169

Special right triangles applications

The 3-4-5 rule to build square corners has been used for years by carpenters to help keep foundations, and buildings square. Using just a saw, and a tape measure, one can create a ninety degree angle, and keep a foundation square. For instance,imagine you are laying a foundation, one can cut a 3, 4, and 5 foot long board. Next, arrange the boards so that they create a triangle in a corner, and a ninety degree angle will be created. This ninety degree angle will keep the foundation square.

Any multiple of 3-4-5 will create a 90 degree angle. Here is a 3-4-5 cheat sheet of the multiples to 80.

3-4-5           15-20-25     27-36-45     39-52-65

6-8-10         18-24-30    30-40-50    42-56-70  

9-12-15        21-28-35    33-44-55     45-60-75

12-16-20      24-32-40   36-48-60     48-64-80