Question

E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F, show that AB X BC = AE X CF

Answer 1

The figure would be something like this..

Answer 2

@imqwerty A lil help, baruto? ;-;

Answer 3

I think we will have to prove the similarity..

Answer 4

Hello hokage :)
Yes you're correct we need to use similarity thing here

Answer 5

But, even if we take the ratio of the sides here.. We wouldn't get the answer. ;-;

Answer 6

On which two triangles should we apply similarity thing
Any ideas?

Answer 7

Don't worry ((:We will get the answer

Answer 8

DEF and FBC?

Answer 9

:D

Answer 10

:D almost there

We need to prove this:
AB x BC =AExCF

We will get BC and CF from triangle FBC but we can't get AE and AB from triangle DEF sooo triangle DEF must be removed and we must consider triangle AEB in place of triangle DEF what do u think? :)
If u get confuzzled anywhere then tell me

Answer 11

I think that could work :)

Answer 12

Yes now try to prove triangle AEB and FBC similar

Answer 13

Well, angle B = angle B (common)
angle EAB = angle BCF (opp angles)
Am I right? ;-;

Answer 14

By AA similarity??

Answer 15

The second one is correct but the first one is not

Answer 16

Hint: think about these- ∠AEB and ∠CBF

Answer 17

Alternate angles? :D

Answer 18

I thought we could take common angles too lol

Answer 19

Yes :D alternate angles

The angle C is not the same in both triangles tho

Answer 20

I seee c:

Answer 21

Now both triangles are similar
ΔABE ∼ ΔCFB (By AA similarity criterion)

\c:/

Answer 22

Taking the ratio of sides, we get,\[\frac{ AE }{ FB } = \frac{ EB }{ BC } = \frac{ AB }{ CF }\]

Answer 23

If we cross multiply, I don't think we'll get the answer :o

Answer 24

We will (:
You wrote the ratio wrong tho
Draw separate figures of the two triangles on a paper and see the corresponding sides and find correct ratios

Answer 25

Remember that we have this-
ΔABE ∼ ΔCFB

Answer 26

Oh, I considered AEB and FCB xD

Answer 27

Then we would get \[\frac{ AB }{ CF } = \frac{ BE }{ FB } = \frac{ AE }{ CB}\]

Answer 28

(: correct

Answer 29

Aha! We could use the first and the last ratios.. And cross multiply.. And bam, we got the answer! Thanks, Baruto! :)

Answer 30

Np hokage (;

Answer 31

:D

Answer 32

:D