Geometry: Special Triangles

On this page we hope to clear up problems that you might have with special triangles, such as a 30°-60°-90°, and theorems that apply to them, such as the Pythagorean Theorem. Read on or follow any of the links below to start better understanding special triangles.

Pythagorean Theorem
45-45-90
30-60-90
Trigonometric ratios
Story problems
Quiz on special triangles

Pythagorean Theorem

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: a^2 + b^2 = c^2, where c represents the hypotenuse.

With this theorem, if you are given the measure of two sides of a triangle, you can easily find the measure of the other side.

Example:

1.  Problem: Find the value of c in the accompanying figure.
   Solution: a^2 + b^2 = c^2         Write the Pythagorean
                                     Theorem and then plug in
                                     any given information.

             5^2 + 12^2 = c^2        The information that was
                                     given in the figure was
                                     plugged in.

             169 = c^2               Solve for c.
             c = 13

45-45-90 Triangles

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.

In a 45-45-90 triangle, the measure of the hypotenuse is equal to the measure of a leg multiplied by SQRT(2).

The accompanying figure presents the theorem in a graphical terms.

30-60-90 Triangles

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.

In a 30-60-90 triangle, the measure of the hypotenuse is two times that of the leg opposite the 30° angle. The measure of the other leg is SQRT(3) times that of the leg opposite the 30° angle.

The accompanying figure presents the theorem in graphical terms.

Trigonometric Ratios

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. There is also an accompanying figure.

sine of angle A = (measure of opposite leg)/(measure of hypotenuse). In the figure, the sin of angle A = (a/c).

cosine of angle A = (measure of adjacent leg)/(measure of hypotenuse). In the figure, the cos of angle A = (b/c).

tangent of angle A = (measure of opposite leg)/(measure of adjacent leg). In the figure, the tan of angle A = (a/b).

Example:

1.  Problem: Find sin A, cos A, and tan A in the
             accompanying figure.
   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.

Story Problems

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.  This figure
             is 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 x^2 = (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
             TI-83 screen: sin^(-1)(10/12) = 56.44  Of course, on the
                                                    TI-82 and TI-83, the
                                                    -1 is super-
                                                    scripted, but text
                                                    only browsers do not
                                                    allow for super-
                                                    scripted characters.
 
             Trigonometric Ratios Table:
             sin 56° = 0.8290
             sin 57° = 0.8387

             By either answer, after rounding to the nearest degree, the
             answer is 56°.

Take the quiz on special triangles. The quiz is very useful for either review or to see if you've really got the topic down.

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