Question

Geometry question: I will post the question in an image file.

Answer 1

I'm not sure how to approach this problem. Ratios or proportions?

Answer 2

for length BC, you may use law of sines in triangles DBE and EBC

Answer 3

Okay, please give me a moment to compute.

Answer 4

In \(\triangle DBE\) apply law of sines\[\dfrac{12}{\sin \angle BED} = \dfrac{4}{\sin \angle DBE}\]

In \(\triangle EBC\) apply law of sines\[\dfrac{BC}{\sin \angle BEC} = \dfrac{5}{\sin \angle EBC}\]

Answer 5

Notice we're given \(\angle DBE \cong \angle EBC \)
and also we have \(\sin \angle BED = \sin\angle BEC\) (why ? )

so we get :
\[\dfrac{12}{BC} = \dfrac{4}{5}\]

solve \(BC\)

Answer 6

sin<BED = sin<BEC becase <BEC = 180degress - <BED, and sinx = sin(180degrees - x) because on the unit circle the right triangle with angle x and angle 180degrees - x has the same reference angle and the same height of sin(x).

Answer 7

BC = 15 :)

Answer 8

Exactly! another way to look at it is
```
sin(180-x) = sin180cosx - cos180sinx
= 0*cosx - (-1)sinx
= sinx
```

Answer 9

i hope you can find other lengths :)

Answer 10

I think I can take it from here! Thanks for your help!!! @ganeshie8

Answer 11

sounds good ;) finding BE length can be tricky.. just holler if you get stuck !

Answer 12

Hi @ganeshie8, what did you get for BE and AD? Using law of sines, cosines, and a system of equations, I calculated the length BE = 4sqrt(10), and the length AD = 4(3sqrt(10) + 5)/23. Just checking to see if anyone else is getting the same thing, thanks! -ceboski

Answer 13

You need the "triangle angle bisector theorem."

Answer 14

The angle bisector of an angle of a triangle divides the opposite side into two segments whose lengths are proportional to their respective adjacent sides.

Answer 15

For the figure above, this theorem means that the following proportion is true:
Segment AD bisects angle BAC.
Then,
\(\dfrac{BD}{AB} = \dfrac{CD}{AC}\)

Answer 16

Using the theorem, you can find BC.
Use triangle DBC.
Then
\(\dfrac{DE}{BD} = \dfrac{EC}{BC} \)
The only unknown is BC, so you can find it.

Answer 17

@mathstudent55 , But BC is not the only unknown! BE, and AD are also unknown. :( Thanks for the tip! Using Triangle Angle Bisector Theorem as you suggested, I find 4/12 = 5/BC, which means BC = 15.

Answer 18

But is it possible to use the triangle angle bisector theorem to find unknowns BE and AD?

Answer 19

I don't see how. I can use the triangle angle bisector theorem only to find BC.

Answer 20

Hi everyone, I got the answer using Stewart's Theorem and the Pythagorean Theorem. BC = 15, BE = 4sqrt(10), AD = 4, and AB = 4sqrt(10). Thanks @mathstudent55, @ganeshie8 and to everyone who participated in this problem! -@cebroski