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RIGHT TRIANGLES WITH SPECIAL ANGLES AND SIDE RATIOS Three types of right triangles are especially significant because of ‘their frequent occurrence. These are the 30°60°90° triangle, the 45°90° triangle, and the 345 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 19lo.30°60°90" triangle. The sine ratio for the 30° angle in figure 1910 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 to tables or to the rule of Pythagoras, find the following lengths and angles in figure 1911: 1. Length of AC. Figure 1911.Finding parts of 30°60°90° triangles. Answers: THE 45°90° TRIANGLE Figure 1912 illustrates a triangle in which two angles measure 45° and the third angle Figure 1912.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 it has 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 to tables or to the rule of Pythagoras, find the following lengths and angles. in figure 1913: 1. AB Answers: 1. 2 
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