Convex quadrilateral ABCD has sides AB=BC=21, CD=15 and AD=9. Given additionally that ∠ABC=60∘, what is the length of BD?

Answer 1

Join the diagonals
Since B - 60° and AB = BC = 21
ABC must be equilateral and AC = 21

Now in triangle ACD we have 3 sides, and by the cosine law I found angle D to be 120°
interesting!
using the sine law in triangle ACD
sin CAD/15 = sin60/21
sin CAD = .61858..
angle CAD = 38.3232... (I stored that)
angle BAD = 60 + 38.32... = 98.21..°

by the cosine law:
BD^2 = 21^2 + 9^2 - 2(21)(9)cos BAD
= 528
BD = √528 or appr 22.98

Answer 2

Reiny in the last step u have done an error as 21^2 =441 + 9^2 = 81 = 522
and 2*21*9*cos(98) = - 54.6
so 522 - (-54.6)=522+52.6=574.6
and square root of 574.6 = 23.97 ~ 24