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.
5-12-13
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.
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
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.
3
5
4
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.
