# Geometry

posted by stranger

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

1. Reiny

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

2. Finalhero

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

## Similar Questions

1. ### maths

ABCD is a convex cyclic quadrilateral such that AB=AD and ∠BAD=90∘. E is the foot of the perpendicular from A to BC, and F is the foot of the perpendicular from A to DC. If AE+AF=16, what is the area of ABCD?
2. ### maths

ABCD is a convex cyclic quadrilateral such that AB=AD and ∠BAD=90∘. E is the foot of the perpendicular from A to BC, and F is the foot of the perpendicular from A to DC. If AE+AF=16, what is the area of ABCD?
3. ### geometry(triangle)

Triangle ABC is a right triangle with ∠ABC=90∘. P is a point within triangle ABC such that ∠APB=∠BPC=∠CPA=120∘. If PA=15 and PB=6, what is the value of PC?
4. ### geometry

ABCD is a convex quadrilateral satisfying AB=BC=CD,AD=DB and ∠BAD=75∘. What is the measure of ∠BCD?
5. ### geometry

ABCD is a quadrilateral with ∠ADC=∠ACD, ∠ACB=∠ABC and CD=7. If triangles ADC and ABC have perimeter 51 and 59, respectively, what is the value of BC?
6. ### geometry...TRIANGLE

ABCD is a quadrilateral with ∠ADC=∠ACD, ∠ACB=∠ABC and CD=7. If triangles ADC and ABC have perimeter 51 and 59, respectively, what is the value of BC?
7. ### Math (Geometry)

ABCD is a parallelogram with side lengths AB=15,BC=10 and ∠ADC=75∘. A point P in the interior of the parallelogram is chosen such that ∠APD+∠CPB=180∘. What is the measure (in degrees) of ∠PAD+∠PCB?
8. ### maths

According to the Journal of Irreproducible Results, any obtuse angle is a right angle! x A P B C D Here is their argument. Given the obtuse angle x, we make a quadrilateral ABCD with ∠DAB = x, and ∠ABC = 90◦ , and …
9. ### maths

According to the Journal of Irreproducible Results, any obtuse angle is a right angle! x A P B C D Here is their argument. Given the obtuse angle x, we make a quadrilateral ABCD with ∠DAB = x, and ∠ABC = 90◦ , and …
10. ### maths

According to the Journal of Irreproducible Results, any obtuse angle is a right angle! x A P B C D Here is their argument. Given the obtuse angle x, we make a quadrilateral ABCD with ∠DAB = x, and ∠ABC = 90◦ , and …

More Similar Questions