Question

An obtuse triangle with area 12 has two sides of lengths 4 and 10. Find the length of the third side. (There are two answers.) (Use law of cosines)

Answer 1

ok. What's the law of cosines?

Answer 2

c^2 = a^2 + b^2 - 2abCosC

Answer 3

right. and what's a formula for the area of the triangle?

Answer 4

...preferably a formula using the variables in the law of cosines formula you just wrote down

Answer 5

(1/2)(B)(H)

Answer 6

We want a formula using a,b,c and C.

Answer 7

Draw yourself a diagram.

Answer 8

idk, that's y I need help

Answer 9

one sec. making one

Answer 10

I don't understand how u got 26cosC

Answer 11

2b. not 6, b, the letter b.

Answer 12

James - It is sinG = h/4, right?

Answer 13

let me fix that.

Answer 14

I found the angle C to be 53.13 degrees

Answer 15

?

Answer 16

Now, use this and the fact that Area = bh/2 to find an expression for sin C, and from that find an expression for cos C. Do not solve for C; there's no need to.

Once you have an expression for cos C, you can then use the law of cosines to find little c, the side length.

Answer 17

I'm confused

Answer 18

Ok. Agree with my new diagram?

Answer 19

talk to me or I'm out of here

Answer 20

I'm here

Answer 21

yea, I get that part

Answer 22

what's next

Answer 23

Ok, now Area = bh/2. Use that relationship to find the value of sin C.

From that, find the value of cos C.

And then you can use the law of cosines to find little c.

Answer 24

b in bh/2 is 10, right?

Answer 25

?

Answer 26

Yes

Answer 27

I didn't get the answer. I got C = 36.9 degrees. What's next??

Answer 28

No, stop, wait. What expression do you get for sin C?

Answer 29

ok, I got 12= (1/2)(10)(4sinC), 12 = 20sinC, sinC = 12/20

Answer 30

yes, sin C = 3/5. Hence what is cos C. Notice it has two values

Answer 31

4/5

Answer 32

?

Answer 33

sin^2 C + cos^2 C = 1. Hence cos C has two values:
\[ \cos C = \pm \sqrt{1-\sin^2 C} \]

Answer 34

so 4/5 = +/- sqrt of 1-sin^2C???

Answer 35

\[ \cos C = \pm \frac{4}{5} \]

Answer 36

Now use the law of cosines to find little c.

Answer 37

\[(4^2) +(10^2)- (2*4*10)(4/5) = c^2 \]

Answer 38

it didn't work, doesn't match the answer key

Answer 39

I know, but isn't that how u find the value for one c?

Answer 40

Nooo.... cos C has two values. And in any case, you need to take the square root to find c.

Answer 41

yes. Did you take the square root?

Answer 42

show me how u do it then

Answer 43

No ... you nut it out. I've already done 90% of this problem for you. You finish it and if it takes you 20 minutes, that's fine. That's how you learn. Figure it out. That's how all of us have to learn. It's painful. But you'll feel better with yourself if you finish it properly.

Answer 44

I don't get how to do it, u told me to use the law of cosines, I can't find the answer

Answer 45

nvm, I got it, thanks!

Answer 46

see? :-)

Now, do yourself a favor. Take a blank piece of paper. Solve the whole problem again. When you can do that without looking at this solution or any other version of the solution, then you understand the question. This is what I used to do a lot of High School, and it helped me really learn the material