Idiot\'s Guides Basic Math and Pre-Algebra

176 Part 3: The Shape of the World

Special Right Triangles

The Pythagorean triples show up a lot when you’re working with right triangles, and certain

families of right triangles tend to show up a lot as well. When an altitude is drawn in an

equilateral triangle, it divides the triangle into two right triangles. Do you remember that this

altitude is one of those super segments that are altitudes, angle bisectors, and medians all in one?

Each of these smaller triangles has a right angle where the altitude meets the base, an angle of

30 r where the altitude bisects the vertex angle, and an angle of 60r. These right triangles are

often called 30-60-90 right triangles because of their angles.

The hypotenuse of the 30r-60r-90r triangle is the side of the original equilateral triangle. The

side opposite the 30r angle is half as large because that altitude was also a median, so it divided

that side into two congruent segments. Using the Pythagorean theorem, you can find the

length of the third side of the right triangle, the side that actually is the altitude. Suppose the

hypotenuse is 1 foot long. That means the leg you know is half of that, or^12 foot.

hh

30 30

60 60

(^1) ⁄ 2 h (^1) ⁄ 2 h
abc
b
b
b
b
22 2
2
22
2
2
1
2
1
1
4
1
3
4
3
4
3
4
3
2
1
2
3