Trig thing! Help please!?

Question

Hi! I have a trigonometry question that has been driving me crazy because I can't figure it out! I've tried for nearly 3 hours to no avail! The problem says:

In the figure, angle B is a right angle and the line segment AD bisects angle A. If AB = 10 and BD = 2, find the exact value of CD.

Now I know that this has to do with half angle/double angle identities, but I'm not sure what else to do at this point. I drew the figure on my Paint program on the computer and I uploaded it to photobucket so you can get a visual of what it looks like from my math book. :)

Anyway, I've tried EVERYTHING!!! But I still can't get it! I would copy over all my work here so you can see what I did and where I went wrong, but it took up 17 pages of paper, so that would be too much for me to transcribe. Can you PLEASE PLEASE PLEASE help with this?!?! I'm desperate! Thank you so much!

Here's the link to my drawing:

http://i223.photobucket.com/albums/dd205/OkayThanksToYou/trigproblem.jpg

Answer 1

You start with what you know:
tan x = 2/10

Angle A is 2x, since you said that the line is dividing the angle in half.

The formula for tan (2x) = (2tan x)/(1-(tan x)^2)
Since tan x = 2/10 (see start of the answer) we get:
tan (2x) = (2*2/10) / (1-(2/10)^2)
tan (2x) = (4/10) / (96/100) = 10/24 = 5/12

One the other hand, tan (2x) = (BC) / (AB) where BC = BD + CD
Thus tan (2x) = (2 + CD) / 10

Now we set the two tan (2x) equal:
5/12 = (2 + CD) / 10
12(2+CD) = 50
24 + 12CD = 50
CD = 26/12 = 13/6

Answer 2

I'll let a be the half angle of A

cos(a) = 2 / 10 = 1 / 5
a = arctan(1/5)

A = 2a = 2arctan(1/5)

tan(A) = (CD + 2) / 10

10 * tan( 2arctan( 1/5 ) ) - 2 = CD
CD = 2.16666667 = 13 / 6

Answer 3

I make the angle BAD = x
This makes the angle BAC = 2x

tan(x) = BD/AB = 0.2

then x = arctan(0.2) = 11.31 degrees = 0.197396 radian (rounded)
and 2x = BC/AB = 22.62 degrees = 0.394791 radian

tan(2x) = .416666...
this makes BC = 4.166666 (4 + 1/6)

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The hard way.

We have tan x = 0.2, we seek tan(2x)
tan(2x) = tan(x+x) = (tanx + tanx) / (1 - tanx tanx)
tan 2x = (0.2 + 0.2) / (1 - 0.04) = 0.4166666...

QFD

Answer 4

if a line bisects the angle it bisects the side...causing it to be the same as BD...or 2.

Answer 5

CD = 13/6
-----------
Ideas:
Let CD = x
CD/AC = BD/AB
x/√[10^2+(x+2)^2] = 2/10
Solve for x by using quadratic formula,
6x^2-x-26 = 0
x = 13/6

Answer 6

tan A/2 = 2/10
A/2 = 11.31 deg
A = 22.62 deg

BC = 10 tan 22.62 = 4.166666
CD = 4.1666 - 2 = 2.16666