Triangle Area Ratios and Point Positioning

[songoku]

Homework Statement

P is a point on triangle ABC such that area PAB : area PBC : area PCA = 2 : 3 : 5
a. Find AQ : QC
b. Show that BP = 0.5 BQ

Homework Equations

The Attempt at a Solution

a. i found that AQ : QC = area QAB : area QBC, but i don't know how to continue ( the answer is 2 : 3)

b. sorry, I'm clueless...

thx

 

[HallsofIvy]

Homework Statement

P is a point on triangle ABC such that area PAB : area PBC : area PCA = 2 : 3 : 5

You mean in triangle ABC don't you?

a. Find AQ : QC

Q? Where did Q come from? Do you mean P?

b. Show that BP = 0.5 BQ

Homework Equations

The Attempt at a Solution

a. i found that AQ : QC = area QAB : area QBC, but i don't know how to continue ( the answer is 2 : 3)

b. sorry, I'm clueless...

thx

 

[songoku]

oh sorry

yes P is a point inside the triangle and the straight line BP intersects the side AC at point Q.

thx

 

[harimakenji]

i tried and got stuck at the same problem...

anyone can help?

 

[snipez90]

Hmm, I think the trick here is to keep using that fact that gave you AQ : QC = area QAB : area QBC, that is, for triangles with the same altitude but different bases, the ratio of their areas is just the ratio of their bases. The corresponding statement for same bases but different altitudes is also useful. I thought I had a solution but I made some pretty strong assumptions and have forgotten all about high school geometry, sorry I can't help further.

 

[kof9595995]

Hmm, I think the trick here is to keep using that fact that gave you AQ : QC = area QAB : area QBC, that is, for triangles with the same altitude but different bases, the ratio of their areas is just the ratio of their bases. The corresponding statement for same bases but different altitudes is also useful. I thought I had a solution but I made some pretty strong assumptions and have forgotten all about high school geometry, sorry I can't help further.

Generally snipez90 is correct, but I can't explain it well without a diagram, I‘ll try to make a picture later.
Hint:Try constructing similar triangles

 

[kof9595995]

Here it is.

 

[songoku]

i think the term 'altitude' is the line that perpendicular to the opposite side.

Is BQ perpendicular to AC ?

thx

 

[kof9595995]

i think the term 'altitude' is the line that perpendicular to the opposite side.

Is BQ perpendicular to AC ?

thx

No, BQ is not necessarily perpendicular to AC

 

[songoku]

nice work

thx a lot all

 

[Leong]

Coordinate geometry

 

[Leong]

Words have spreaded

 

[songoku]

wow, you use geometry coordinate again to solve triangle problem.

awesome !

thx a lot