Parallel Lines        Congruent Tri.        Congruent R. Tri.        Isosc. and Equil.        Quadrilaterals        Parallelograms        Ratios        Similar Polygons    Special Triangles       Circles        Area        Coordinate Geo.        Triangle Ineq.        Solids        Computer Fun

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. Pythagorean Theorem 45-45-90 30-60-60 Trigonometric ratios Story problems 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. With this theorem, if you are given the measures of two sides of a triangle, you can easily find the measure of the other side. 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 Back to Top  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 following figure presents the theorem in graphical terms. Back to Top  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 30o angle.  The measure of the other leg is SQRT(3) times that of the leg opposite the 30o angle. The following figure presents the theorem in graphical terms. Back to Top  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). 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). 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. Back to Top  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. Back to Top  Take the Quiz on special triangles.  (Very useful to review or to see if you've really got this topic down.)  Do it!