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.
2. Size of angle A.
3. Size of angle B.
4. Length of RT.
5. Length of RS.
6. Size of angle T.
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
2. BC
3. Angle B
Answers:
1. 2
2. 2
3. 45°
