In the shape below, the length of BC is 3.1cm and the length of CD is 5.9cm.

Question

a) Calculate the size of angle BDC.
b) The length of AC is twice the length of BC. Calculate the size of angle ADC.
Give each of your answers in degrees to 1 d.p.

Answer 1

To calculate angle BDC, we first need to find the length of BD using the Pythagorean theorem.

BD^2 = BC^2 + CD^2
BD^2 = 3.1^2 + 5.9^2
BD^2 = 9.61 + 34.81
BD^2 = 44.42
BD ≈ √44.42
BD ≈ 6.66 cm

Now we can calculate angle BDC using the cosine rule:
cos(BDC) = (BD^2 + CD^2 - BC^2) / (2 * BD * CD)
cos(BDC) = (6.66^2 + 5.9^2 - 3.1^2) / (2 * 6.66 * 5.9)
cos(BDC) = (44.36 + 34.81 - 9.61) / (78.54)
cos(BDC) = 69.56 / 78.54
cos(BDC) ≈ 0.885
BDC ≈ cos^(-1)(0.885)
BDC ≈ 28.5 degrees

Next, we calculate the length of AC:
AC = 2 * BC
AC = 2 * 3.1
AC = 6.2 cm

Now we can calculate angle ADC using the cosine rule:
cos(ADC) = (AC^2 + CD^2 - AD^2) / (2 * AC * CD)
cos(ADC) = (6.2^2 + 5.9^2 - 6.66^2) / (2 * 6.2 * 5.9)
cos(ADC) = (38.44 + 34.81 - 44.36) / (23.24)
cos(ADC) = 28.89 / 23.24
cos(ADC) ≈ 1.244
ADC ≈ cos^(-1)(1.244)
ADC ≈ ERROR (Impossible triangle, since cos^(-1) of a value greater than 1 is not defined)

Therefore, the size of angle BDC is 28.5 degrees, and the size of angle ADC cannot be calculated as it would result in an error due to an impossible triangle.