In △ A B C, side BC is divided by D in a ratio of...

Ishaan Booker

Ishaan Booker

Answered

2022-07-16

In A B C, side BC is divided by D in a ratio of 5 to 2 and BA is divided by E in a ratio of 3 to 4 as shown in Figure. Find the ration in which F divides the cevians AD and CE i.e. find E F : F C and D F : F A.

Answer & Explanation

yermarvg

yermarvg

Expert

2022-07-17Added 19 answers

Step 1
A general method to solving these kinds of things is through the Ratio Lemma, which is often applied in high school olympiad geometry. It states that if AD is a cevian in A B C, then B D D C = A B A C sin B A D sin C A D . The proof of this is quite simple; just apply the sine law to triangles ABD and CAD.
Step 2
So for this problem, the ratios sin B A D : sin C A D is clearly equal to sin E A F : sin C A F. So E F : F C is equal to ( B D / D C ) ( A E / A B ) = 10 / 7. D F : F A is calculated similarly.
The Ratio Lemma can also be used to prove many basic facts in projective geometry; e.g. the invariance of the cross-ratio when projected through a point onto a line.
Levi Rasmussen

Levi Rasmussen

Expert

2022-07-18Added 6 answers

Step 1
Use mass points. We assign B a mass of 4. So A has a mass of 3 and C has a mass of 10. E has a mass of 7 and D has a mass of 14.
Step 2
So we have A F F D = 14 3
then E F F C = 10 7

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?