On this page, we hope to clear up problems that you might
have with special triangles, such as a
30o-60o-90o, and theorems
that apply to them, such as the Pythagorean Theorem.
Scroll down or click on one of the links below to start better
understanding special triangles.
Quiz on Special Triangles
One of the most famous mathematicians who has ever lived,
Pythagoras, a Greek scholar who lived way back in the
6th century B.C. (back when Bob Dole was learning geometry), came
up with one of the most famous theorems ever, the Pythagorean Theorem.
It says - in a right triangle, the square of the measure of the hypotenuse equals
the sum of the squares of the measures of the two legs.
This theorem is normally represented by the following equation:
a2 + b2 = c2, where
c represents the hypotenuse.
1. Problem: Find the value of c. Solution: a2 + b2 = c2 Write the Pythagorean Theorem and then plug in any given information. 52 + 122 = c2 The information that was given in the figure was plugged in. 169 = c2 Solve for c c = 13
One of the special right triangles which we deal with in geometry
is an isosceles right triangle. These triangles are also
known as 45-45-90 triangles (so named because
of the measures of their angles). There is one theorem that
applies to these triangles. It is stated below.
There's another kind of special right triangle which we deal
with all the time. These triangles are known as
30-60-90 triangles (so named because of the measures
of their angles). There is one theorem that applies to
these triangles. It is stated below.
While the word trigonometry strikes fear into the hearts of many, we made it through (amazing as it may seem to us), and hope to help you through it, too! Each of the three basic trigonometric ratios are shown below.
sine of angle A = (measure of
opposite leg)/(measure of hypotenuse). In the figure, the
sin of angle A = (a/c).
1. Problem: Find sin A, cos A, and tan A. Solution: sine = (opposite/hypotenuse) sine = 5/13 cosine = (adjacent/hypotenuse) cos = 12/13 tangent = (opposite/adjacent) tan = 5/12
Be aware that, although the example above seems to indicate otherwise, the values for the trigonometric ratios depend on the measure of the angle, not the measures of the triangle's sides.
Many problems ask that you find the measure of an angle or a segment that cannot easily be measured. Problems of this kind can often be solved by the application of trigonometry. Below is an example problem of this type.
1. Problem: A ladder 12 meters long leans against a building. It rests on the wall at a point 10 meters above the ground. Find the angle the ladder makes with the ground. Solution: Make sure you know what is being asked. Then use the given information to draw and label a figure. Here's our idea of a figure for this problem: Choose a variable to represent the measure of the angle you are asked to find. Using the variable you have chosen, write an equation that will solve the problem. sin x2 = (10/12) The above equation is derived from the given information and the knowledge of the sine ratio. Find the solution using a calculator's Arcsine function or a table of trigonometric ratios. TI-82 screen: sin-1 (10/12) = 56.44 Trigonometric Ratios Table: sin 56o = 0.8290 sin 57o = 0.8387 By either answer, after rounding to the nearest degree, the answer is 56o.
Take the Quiz on special triangles. (Very useful to review or to see if you've really got this topic down.) Do it!