Question

"Consider the triangle ABC where BÂC = 70°, AB = 8cm and AC = 7cm. The point D on the side BC is such that BD/DC = 2. Determine the length of AD."
I've done this about 10 times now and got roughly 6.32cm everytime, while the right correct asnwer should be 6.12cm. Any suggestions?

Answer 1

Could you draw out the figure for us first? :)

Answer 2

Sure :D

Answer 3

Does this look like it?

Answer 4

Exactly like that, and waaay better than what I was drawing :P

Answer 5

Using the sine and cosine rules, I keep getting approx. 6.33 for AD, whereas it should be around 6.12 cm.

Answer 6

Since for triangle ABC you have 2 sides and an angle, you can use the law of cosines to find side AC. Then solve for angle C.
Then with 2:1 ratio, you can find DC. Using angle C, DC and AC you can find AD.

Answer 7

Yes that is exactly what I did, yet my answer was still off about 0.2cm (without roudning). I was hoping if someone could do the calculations as well and verify whether my answer is correct or not.

Answer 8

I get BC = 8.643 cm

Answer 9

Yes I had that too

Answer 10

So far so good

Answer 11

DC = 8.643 cm/3 = 2.881 cm

Answer 12

m<C = 60.439 deg

Answer 13

Yes I'm getting the same figures as you

Answer 14

This gives m<ADC = 95.4 deg

Answer 15

AD = 6.12 cm

Answer 16

Normally I would assume my answer is correct, but the weird thing is that this question is filed under the title "Complex numbers", as are all the complex-numbered questions before it, so I feel like I might be doing something wrong here since I haven't used any "i"s.

Answer 17

There are no complex numbers here. This is just a normal triangle that can be solved since we are given enough information.

Answer 18

I calculated AD using the sine rule without calculating ADC-angle, so would that be the problem?

Answer 19

What exactly did you use to calculate AD using the sine rule? I mean whcih sides and angles did you use?

Answer 20

I assumed that the angle DAC is 70/3. Is that the problem? Besides that I used the side DC, and angle ACB

Answer 21

so my equation was \[\frac{ \sin(DAC) }{ DC }=\frac{ \sin(ACB) }{ AD }\]

Answer 22

You can't assume that. In fact angle DAC turns out to be 24.2 deg which is not m<BAC/3.

Answer 23

Aaa I see. Can you also explain why that cannot be assumed? Since D cuts BC in a 2:1 ratio, I thought the line AD would divide the angle similarly.

Answer 24

And how did you calculate the angle ADC?

Answer 25

The only thing I know realetd to an angle about a triangle's side being cut into segments in a proportional way is when an angle bisector of a triangle is drawn to the opposite sides, the lengths of the segments are in proportion to the lengths of the adjacent sides.

Answer 26

Yeah now that I think about it it does make more sense that way

Answer 27

Now for angle ADC, let's just look at what we need:

Answer 28

From here you can get AD directly just like we got BC directly in triangle ABC.

Answer 29

By the law of cosines, you can get AD = 6.12 cm directly.

Answer 30

Aaaa right, I totally missed that for some reason.

Answer 31

Thank you so much, your explanations were very detailed and a lot more than what I expected I'd get!

Answer 32

You're welcome.