PRACTICE ANOTHER In the figure, triangle ABC is a right triangle,

Question

CQ = 6, and BQ = 8. Also, ∠AQC = 30° and ∠CQB = 45°. Find the length of AQ. [Hint: First use the Law of Cosines to find expressions for a2, b2, and c2.]

AQ=

Answer 1

Right triangle ABC where C is the right angle...

Q is on side AB

Then angle QCB=60 and angle CBQ is 75

Let x = CB = a

Per law of cosines:

x^2 = 6^2 + 8^2 - 2(6)(8) cos 45

= 100 - 48*sqrt(2)

So then x is approximately 5.6672523330174242081448041317437

angle CAB must be 15

let y = AQ

law of sines says:

sin 30/y = sin 15/6

cross multiplies: 6*sin 30 = y * sin 15

y = 3/sin 15

which is approximately 11.591109915468819440996918396747

The hypotenuse is then 19.591109915468819440996918396747

Pythagorean says that AC is 18.753502039515587954646705108132