Convex quadrilateral ABCD has sides AB=BC=21, CD=15 and AD=9. Given additionally that ∠ABC=60∘, what is the length of BD?
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
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
To find the length of BD in the convex quadrilateral ABCD, we can use the law of cosines. The law of cosines states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of the two sides and the cosine of the included angle.
In the given information, we can see that triangle ABC is an equilateral triangle with side length 21. So, all angles of triangle ABC are 60 degrees. Thus, angle BCA is also 60 degrees.
Let's label angle BCD as θ.
Using the law of cosines in triangle BCD, we can write:
BD^2 = BC^2 + CD^2 - 2 * BC * CD * cos(θ)
Substituting the given values:
BD^2 = 21^2 + 15^2 - 2 * 21 * 15 * cos(θ)
We need to find the value of cos(θ). To do that, we can use the information that sides AB, BC, and CD are all equal. This implies that angle BCD is equal to angle BAC.
Angle BAC = 180 - ∠ABC = 180 - 60 = 120 degrees.
Since angle BAC is equal to angle BCD, we can conclude that θ = 120 degrees.
Now, we can substitute the value of θ in the equation for BD:
BD^2 = 21^2 + 15^2 - 2 * 21 * 15 * cos(120)
To evaluate cos(120), we can use the fact that cos(120) = -cos(60) = -0.5.
BD^2 = 21^2 + 15^2 - 2 * 21 * 15 * (-0.5)
Simplifying:
BD^2 = 441 + 225 + 630
BD^2 = 1296
Taking the square root of both sides:
BD = √1296
BD = 36
Therefore, the length of BD is 36 units.