Answer:
AC ≈ 9.84
BD ≈ 15.32
Step-by-step explanation:
You want the lengths of diagonals AC and BD in kite ABCD with AB=AD=10 and A=100°, D=64°.
Law of sines
The desired diagonal lengths can be found using the law of sines.
AC/sin(D) = AD/sin(∠ACD)
To make use of this, we need to know the measure of ∠ACD. We know that AC bisects angle A, so ...
∠ACD = 180° -(A/2) -D
∠ACD = 180° -50° -64° = 66°
Now, we can write ...
AC = AD·sin(D)/sin(66°) = 10·sin(64°)/sin(66°) ≈ 9.84
The length of AC is about 9.84 units.
In like fashion, ...
BD/sin(A) = AB/sin(∠ADB)
The measure of ∠ADB can be found using the fact that ∆ABD is isosceles.
2·∠ADB +A = 180°
∠ABD = (180° -A)/2 = 90° -(A/2) = 40°
Now, we can write ...
BD = AB·sin(A)/sin(∠ABD) = 10·sin(100°)/sin(40°) ≈ 15.32
The length of BD is about 15.32 units.
