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Right triangles with special angles and side ratios

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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 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 to tables or to the rule of Pythagoras, find the following lengths and angles in figure 19-11:

1. Length of AC.
2. Size of angle A.
3. Size of angle B.
4. Length of RT.
5. Length of RS.
6. Size of angle T.

Figure 19-11.-Finding parts of 30°-60°-90° triangles.

Answers:

THE 45°-90° TRIANGLE

Figure 19-12 illustrates a triangle in which two 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 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 19-13:

1. AB 
2. BC 
3. Angle B

Answers:

1. 2 
2. 2 
3. 45°




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