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

Since we are given the lengths of BC and CD, we can use the cosine rule to find the angle BDC.

a) Let's denote the angle BDC as x. We can use the cosine rule to find x:

CD² = BC² + BD² - 2 * BC * BD * cos(x)
(5.9)² = (3.1)² + (BD)² - 2 * (3.1) * (BD) * cos(x)
34.81 = 9.61 + BD² - 6.2BD * cos(x)

We can use the cosine rule cosine rule to find the length of BD:
BD = √(BC² + CD² - 2 * BC * CD * cos(∠BDC))
BD = √(3.1² + 5.9² - 2 * 3.1 * 5.9 * cos(x))
BD = √(9.61 + 34.81 - 36.58cos(x))
BD = √44.42 - 36.58cos(x))

Now substitute BD into the equation and solve for cos(x):
34.81 = 9.61 + [√44.42 - 36.58cos(x)]² - 6.2[√44.42 - 36.58cos(x)]cos(x)
34.81 = 9.61 + 44.42 - 2(√44.42)(36.58)cos(x) + 36.58²cos²(x) - 6.2[44.42cos(x) - 36.58cos²(x))
34.81 = 54.03 - 2(√44.42)(36.58)cos(x) + 36.58²cos²(x) - 274.564cos(x) + 23.3736cos²(x)
23.22cos²(x) - 276.484cos(x) + 19.22 = 0
Solving the quadratic gives
cos(x) = 2.7073 or -3.8432
but -1 ≤ cos(x) ≤ 1 and therefore cos(x) = 0.27073

Now finding x,
cos⁻¹(0.27073) = x
x ≈ 74.9°

Therefore, the size of angle BDC is approximately 74.9°.

b) Since AC is twice the length of BC, it means AC = 2*3.1 = 6.2 cm.

Now we can use the cosine rule again to find the angle ADC:

AC² = AD² + CD² - 2 * AD * CD * cos(∠ADC)
6.2² = AD² + 5.9² - 2 * AD * 5.9 * cos(x)
38.44 = AD² + 34.81 - 11.8AD * cos(x)

Now, we already know that cos(x) = 0.27073, we can use this value to find AD:

38.44 = AD² + 34.81 - 11.8AD * 0.27073
38.44 = AD² + 34.81 - 3.19196AD
AD² - 3.19196AD + 3.63 = 0
Solving the quadratic gives
AD = 0.03995 or 3.19196
but AD must be positive, so
AD ≈ 3.19196

Now, we can find the angle ADC:
cos(∠ADC) = (AD² + CD² - AC²) / (2 * AD * CD)
cos(∠ADC) = (3.19196² + 5.9² - 6.2²) / (2 * 3.19196 * 5.9)
cos(∠ADC) = (10.1838 + 34.81 - 38.44) / (11.90296)
cos(∠ADC) = 0.341989
∠ADC = cos⁻¹(0.341989)
∠ADC ≈ 69.60°

Therefore, the size of angle ADC is approximately 69.6°.