The figure below shows a triangular wooden frame ABC. The side AD of the frame has rotted and needs to be replaced: Triangle ABC has length of BC equal to 16 inches. A line segment DC joins the point D on AB with point C. Measure of angle ABC is 90 degrees, ACD is 15 degrees and measure of angle DCB is 30 degrees. What is the length of the wood that is needed to replace AD?

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The figure below shows a triangular wooden frame ABC. The side AD of the frame has rotted and needs to be replaced: Triangle ABC has length of BC equal to 16 inches. A line segment DC joins the point D on AB with point C. Measure of angle ABC is 90 degrees, ACD is 15 degrees and measure of angle DCB is 30 degrees. What is the length of the wood that is needed to replace AD?

Mathematics
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@stacy.reed08
6.8 inches 6.2 inches 3.2 inches 2.8 inches
I thought maybe doing 15 divided by 2 would work but i got 7.5 inches and thats not an option so im assuming there's a step im missing

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Other answers:

I should have looked down at the picture...it took me forever to draw by hand.
ok, so if you think about it, there is a little triangle inside the larger triangle ABC
Yes i see that
ok, let's draw that triangle out separately
|dw:1440107397764:dw|
I take that back...sorry, I thought we were trying to figure out line segment DC...just kidding, we want AD
im guessing in order to find out the rotting piece needed to be replaced we have to find out how long side CD is?
ok, I know the answer and there is probably an easier way to get there, but this is what makes sense in my mind...
use the smaller triangle to figure out the length of DB.
How do we do that
OH
|dw:1440107651617:dw|
ok so BC is 16 inches DB is 30 inches and 180-16-30=134 so Dc is 134
nope, we don't know DB...we know angle DCB is 30 degrees and we know side BC is 16 inches.
but since the angle is 30 degress wouldn't the side across from it be 30 inches?
|dw:1440107798750:dw|
if we pull the little triangle out it's easier to work with
and nope...the side across from it is going to be an ugly number that we'll round a little bit to make it easier to work with.
ok lol
so the missing side x and the angle are related how?
because they're opposite from each other
ok, and how are the angle and the 16 inch side related?
they are adjacent to each other
so you have opposite and adjacent which means what trig function?
Toa= tangent
ok so plug it in and actually solve it out to get a single number. It will be ugly looking so round to 2 digits past the decimal.
How would i solve/calculate this?
Actually one sec
I've gotta grab something off the printer...I'll be back in a minute
Alright
30(tan)16=9?
9.24 just to be a little more accurate
sorry it took so long, I locked my keys in my room and had to go around.
lol it's fine so what would be the answer here then?
the largest option available is 6.8 inches.
or highest rather
ok, so right now you're only part way done...
now let's look at the big triangle...
Lol i just realized
|dw:1440108392044:dw|
so we have the adjacent and opposite again?
yup.
I have to be somewhere at 7 on the dot so in an hour i have to go, just letting you know
And once again i deeply appreciate your patience in helping me solve my geometry problems
So we would do 15(tan)30?
I actually need to leave as soon as we're done with this problem. I'm in grad school and have a paper to write...plus it's past 5 and I'm still at school :(
the angle isn't 30...we need all of angle C
|dw:1440108572263:dw|
ahh so 45, and sorry for taking up so much time lol
you're fine. like I said before, I don't teach math anymore so this is actually fun for me. :) and yes, 45
so what's the length of our missing side?
45?
nope...plug it in just like you did before...your set up was write, just the angle number was wrong. so 16 tan 45
it comes out as 16
yup. so know you know side AB (the entire side) is 16 inches and you know DB is 9.24 so that means AD is ?
or when i put the calculator in Radian form, its 25.9 or 26
oh ok disregard that comment then
the angle was given in degrees so don't use radian form :)
omg 6.76
You are amazing holy christ
i have 3 more answers but with my new found knowledge maybe i can solve them on my own lol
questions not answers*
well I wouldn't go that far...I realized just know that the big triangle ABC is a 45-45-90 triangle which means both legs are equal...
we didn't need to do all that last part....
lol its fine
Do you mind staying for just 3 more questions?
Like just answering them would be fine, as we're both on a time budget, but if you can't then i completely understand.
Unfortunately, I really have to go. The custodian is starting to wonder why I'm still here :)
lol alright see you then thanks for the help and time once again!
no problem! Best of luck on the rest of your problems :)
lol thank you

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