Triangle Area Ratios and Point Positioning

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In summary, the conversation discusses finding the ratio of AQ to QC in a triangle ABC, where P is a point inside the triangle and the straight line BP intersects the side AC at point Q. The solution involves using the fact that triangles with the same altitude but different bases have a ratio of their areas equal to the ratio of their bases. Additionally, constructing similar triangles and using coordinate geometry can also help in solving the problem.
  • #1
songoku
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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
 
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  • #2
songoku said:

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
 
  • #3
oh sorry

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

thx
 
  • #4
i tried and got stuck at the same problem...

anyone can help?
 
  • #5
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.
 
  • #6
snipez90 said:
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
 
  • #7
Here it is.
 

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  • #8
i think the term 'altitude' is the line that perpendicular to the opposite side.

Is BQ perpendicular to AC ?

thx
 
  • #9
songoku said:
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
 
  • #10
nice work

thx a lot all
 
  • #11
Coordinate geometry
 

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  • #12
Words have spreaded
 

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  • #13
wow, you use geometry coordinate again to solve triangle problem.

awesome !

thx a lot
 

1. What is the formula for finding the area of a triangle?

The formula for finding the area of a triangle is A = (base * height) / 2. This formula applies to all types of triangles, whether they are equilateral, isosceles, or scalene.

2. How do you find the missing side of a triangle?

To find the missing side of a triangle, you can use the Pythagorean theorem (a² + b² = c²) if it is a right triangle. For other types of triangles, you can use the law of cosines or the law of sines to find the missing side.

3. Can you have a triangle with sides of different lengths?

Yes, a triangle can have sides of different lengths. This type of triangle is called a scalene triangle. It is the only type of triangle that does not have any equal sides or angles.

4. How many degrees are in a triangle?

A triangle always has a total of 180 degrees, regardless of its size or shape. This is because the three angles inside a triangle must always add up to 180 degrees.

5. Can you have a triangle with all angles measuring 90 degrees?

Yes, a triangle with all angles measuring 90 degrees is called a right triangle. This type of triangle is used in the Pythagorean theorem and has special properties, such as the fact that its sides follow the ratio 3:4:5.

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