RIGHT TRIANGLES WITHSPECIAL ANGLES AND SIDE RATIOS
Three types of right triangles are especiallysignificant because of ‘their frequent occurrence. These are the 30°-60°-90° triangle, the 45°-90° triangle, and the 3-4-5 triangle.
THE 30°-60°-90° TRIANGLE
The 30°-60°-90° triangle is so named because these are the sizes of its three angles.The sides of this triangle are in the ratio of 1 to to 2, as shown in figure 19- 10.
Figure 19-lo.-30°-60°-90" triangle.
The sine ratio for the 30° angle in figure 19-10 establishes the proportionate values of the sides. For example, we know that the sine of 30° is 1/2; therefore side AB must be twice as long as BC. If side BC is 1 unit long, then side AB is 2 units long and, by the rule of Pythagoras, AC is found as follows:
Regardless of the size of the unit, a 30°-60°-90° triangle has a hypotenuse which is 2 times as long as the shortest side. The shortest side is opposite the 30° angle. The side opposite the 60° angle is fl times as long as the shortest side. For example, suppose that the hypotenuse of a 30°-60°-90° triangle is 30 units long; then the shortest side is 15 units long, and the length of the side opposite the 60° angle is 15 units.
Practice problems. Without reference totables or to the rule of Pythagoras, find the following lengths and angles in figure 19-11:
1. Length of AC.
Figure 19-11.-Finding parts of30°-60°-90° triangles.
THE 45°-90° TRIANGLE
Figure 19-12 illustrates a triangle in whichtwo angles measure 45° and the third angle
Figure 19-12.-A 45°-90° triangle.
measures 90°. Since angles A and B are equal, the sides opposite them are also equal. Therefore, AC equals CB. Suppose that CB is 1 unit long; then AC is also 1 unit long, and the length of AB is calculated as follows:
Regardless of the size of the triangle, if ithas two 45° angles and one 90° angle, its sides are in the ratio 1 to 1 to . For example, if sides AC and CB are 3 units long, AB is 3 units long.
Practice problems. Without reference totables or to the rule of Pythagoras, find the following lengths and angles. in figure 19-13: