In a right-angled triangle abc, the angle at is 30 degrees, the sum of the hypotenuse and the smaller
In a right-angled triangle abc, the angle at is 30 degrees, the sum of the hypotenuse and the smaller leg is 27 cm. Find the smaller leg.
In a right-angled triangle ABC it is known:
Angle B = 30 °;
Angle C = 90 °;
Hypotenuse AB + smaller leg AC = 27.
Find the smaller AC leg, that is, the AC leg.
Decision:
1) AB + AC = 27;
sin B = AC / AB (ratio of the opposite leg to the hypotenuse).
AC = AB * sin B;
Substitute AC = AB * sin B into the expression AB + AC = 27 and get an equation with an unknown value of AB.
AB + AB * sin B = 27;
AB * (1 + sin B) = 27;
AB * (1 + sin 30) = 27;
AB * (1 + 1/2) = 27;
AB * 3/2 = 27;
AB / 2 = 9;
AB = 9 * 2;
AB = 18;
2) Now we will find the smaller leg AC and 3 formulas.
AC = AB * sin B = 18 * sin 30 = 18 * 1/2 = 18/2 = 9.
Answer: AC = 9.
