## Tuesday, February 8, 2011

### Isosceles Triangles - calculating area (given sides)

This is an essential skill for the SAT.  It is commonly needed to directly solve a math problem on the SAT or as an important step in a larger problem.

Definition:  Isosceles triangles are triangles that have at least 2 sides of equal length.

There are many ways to find the area of various types of triangles.  In this post, I will show you how to calculate the area of an isosceles triangle given the lengths of the 3 sides.  The method illustrated is relatively straight-forward and is used by many students.

Given: isosceles triangle with 3 given sides.

To find: area of triangle (height is by-product).

Method:

1. Draw an altitude from the vertex between the 2 equal sides to the opposite side.

This opposite side serves as the base corresponding to the altitude just drawn.  The base is now split in half.  The isosceles triangle is split into 2 congruent right triangles (can you prove this?).

2. Next, find the length of this altitude using the Pythagorean theorem.

3. Plug & chug into A = (1/2) b * h.

Example:

A triangle is given with sides 5, 5, 8.  Choose the side of 8 to be the base.  Draw the altitude to this base.  Each of the right triangles formed has a hypotenuse of 5 and one leg of 4 (half of the side of 8).  The other leg is the same as the altitude of the original triangle.  It has length 3 by the Pythagorean theorem (3^2 + 4^2 = 5^2).  Thus, the original triangle has base 8, height 3, and area (1/2) * 8 * 3 = 12.

Note:  Equilateral triangles are isosceles.  You can find their areas in the same way.  The 2 right triangles you construct will be 30-60-90 triangles.  The height of the equilateral triangle will then be x * sqrt(3)/2, where x is the length of one of the sides.  The area of the equilateral triangle is then x^2 * sqrt(3) / 4.  These formulas for the height and area of an equilateral triangle are worth memorizing because equilateral triangles are so common on the SAT.

Practice:

1. Find the height and area of a triangle with sides 13, 13, 10.  (Area is 60.)

2. Find the height and area of a triangle with sides 25, 25, 30.  (Area is 300.)

3. Find the height and area of a triangle with sides 3, 3, 3.  (Area is 9 * sqrt(3) / 4.)