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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 306090 triangle, the 4590 triangle, and the 345 triangle. THE 306090 TRIANGLE The 306090 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.306090" 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 6090 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 306090 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 306090 triangles. Answers: THE 4590 TRIANGLE Figure 1912 illustrates a triangle in which two angles measure 45 and the third angle Figure 1912.A 4590 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 