Find the angle BCD

Question

This problem refers to the figure below, which shows a diagram for a bike frame.

Given BC = 49 cm, BD = 63 cm, CD = 76 cm, ∠ABC = 52°, and ∠ACB = 67°, find the following. (Round your answers to the nearest whole number.)

(b)    ∠BCD

∠BCD = _____ °

Answer 1

There are two triangles in this figure: ABC and BCD. Your question is asking for an interior angle of triangle BCD. We can solve this just by looking at that triangle, and ignoring the other one.

The problem gives us the lengths of all the sides in BCD. Accordingly, we can apply the Law of Cosines to get the angle measurements. The formula works like this: c² = b² + d² - 2bd * cos(C). The lowercase letters refer to the side lengths opposite the angles: d is the side opposite angle D, or BC, b is the site opposite angle B, or CD, etc.

Substituting in the numbers, we get: 63² = 49² + 76² - 2(49)(76)cos(C). Subtracting the two constant terms and dividing by the coefficient of the cosine of C, we get cos(C) = (63² - 49² - 76²) / (-2(49)(76)). So the value of C will be cos^-1((63² - 49² - 76²) / (-2(49)(76))). Plugging that into a calculator, and we get that angle C of triangle BCD (i.e. angle BCD itself) is 55.6 degrees.