Math Help: Radicals in Geometry

Help with Radicals in Geometry

In math, a radical is the root of a number. Examples of radicals include (square root of 4), which equals 2 because 2 x 2 = 4, and (cube root of 8), which also equals 2 because 2 x 2 x 2 = 8. The most common type of radical that you'll use in geometry is the square root.

Pythagorean Theorem

Students typically learn about the Pythagorean Theorem in 8th grade. This theorem is actually a formula that helps you find the length of one side of a right triangle (a triangle with an angle that measures 90 degrees). The theorem states that a^2 + b^2 = c^2, where 'c' is the triangle's hypotenuse (longest side) and 'a' and 'b' are the triangle's other two sides.

Although this theorem doesn't contain a radical, you'll need to use radicals to solve for any variable in the equation. For instance, imagine you're asked to find the length of a triangle's hypotenuse when the lengths of the other 2 sides are 5 centimeters and 12 centimeters, respectively. Here's how you'll solve this problem using radicals:

a^2 + b^2 = c^2

5^2 + 12^2 = c^2

25 + 144 = c^2

169 = c^2

(square root of 169) = (square root of c^2)

c = 13

Special Right Triangles

In high school, when you work with trigonometry, you'll also learn about several types of special right triangles. These include the 30-60-90 right triangle and the 45-45-90 right triangle, which are named after the measures of their angles.

You'll use radicals when you work with these types of triangles because the relationships between their side lengths can be expressed using radicals. For example, in a 30-60-90 right triangle, the ratio for the shorter side (SS), the longer side (LS) and the hypotenuse (H) can be expressed like this: SS:LS:H = 1:(square root of 3):2. For 45-45-90 right triangles, the ratio is 1:1:(square root of 2), where (square root of 2) represents the hypotenuse.

You can use these ratios to identify the length of the other two sides of the triangle when you only know the length of one side. For instance, if one of the non-hypotenuse sides of a 45-45-90 right triangle is 4 centimeters, then the lengths of the other two sides would be 4 and 4(square root of 2) centimeters, respectively. Likewise, if a 30-60-90 right triangle has a longer leg that measures 2 (square root of 3) inches, then the shorter leg would be 2 inches long and the hypotenuse would be 4 inches long.