SOLUTION: This problem just confuses me to no end. Please help. M is the midpoint of AB and P is any point in AM. Line MD meeting BC at D is parallel to PC. Prove that triangle BPD has half

Question 31174: This problem just confuses me to no end. Please help.
M is the midpoint of AB and P is any point in AM. Line MD meeting BC at D is parallel to PC. Prove that triangle BPD has half the area of triangle ABC.
This que
In triangle ABC, stion is so hard. I really hope you can find a solution! Thank you!!!
Answer by venugopalramana(3286) (Show Source): You can put this solution on YOUR website!
This problem just confuses me to no end. Please help.
In triangle ABC, M is the midpoint of AB and P is any point in AM. Line MD meeting BC at D is parallel to PC. Prove that triangle BPD has half the area of triangle ABC.
OK ...P IS ANY POINT ON AM.LET
MP=X*MA=X*AB/2..SINCE M IS MIDPOINT OF AB.
IN TRIANGLE BPC,WE HAVE MD || PC.HENCE
BD/DC=BM/MP=(AB/2)/(X*AB/2)=1/X
BD/DC=1/X.....OR....BD/(BD+DC)=1/(1+X)
BD/BC=1/(1+X)...OR.....BD=BC/(1+X)................................I
SIMILARLY FROM BM/MP = 1/X.....OR.......BM/(BM+MP)=1(1+X)
BM/BP=1/(1+X)
(AB/2)/BP=1/(1+X)......OR.....BP=AB*(1+X)/2.....................II
NOW AREA OF TRIANGLE = (1/2)*BASE*ALTITUDE
AREA OF TRIANGLE ABC = A(ABC)SAY =(1/2)BC*AB*SIN(B)..................III
SINCE ALTITUDE =AB*SIN(B).......
SIMILARLY.....
AREA OF TRIANGLE BPD=A(BPD) SAY =(1/2)BD*BP*SIN(B)........................IV
DIVIDING EQN.III/EQN.IV.....WE GET.....
A(ABC)/A(BPD)=BC*AB/BD*BP..............................................V
BUT BD=BC/(1+X) AND BP =AB*(1+X)/2...SO
BD*BP=BC*AB*(1+X)/2(1+X)=BC*AB/2...SUBSTITUTING IN EQN.V WE GET
A(ABC)/A(BPD)=BC*AB/(BC*AB/2)=2.....PROVED
RELATED QUESTIONS
Hello, Could someone please help me with this question? I have a triangle with A(0,0),... (answered by stanbon)

Let ABC be a triangle such that AB= 6cm, AC=8cm and BC 10cm and M be the midpoint of BC... (answered by KMST)

In triangle ABC, M is the midpoint of line AB. Let D be the point on line BC such that... (answered by math_helper)

This prove is really confusing me and below have provided what i think i understand but i (answered by drk)

Could you please help me solve this problem? Prove the Median Inequality: If M is the... (answered by richard1234,Edwin McCravy)

Please help me solve this question: ABCD is a square with 8 inch sides. Point E is the... (answered by macston)

In an isosceles triangle labelled ABC, with BC as the base, and AB as the hypotenuse, and (answered by greenestamps)

Please help me tutors Find the coordinate of point B in the segment AB if M is the... (answered by drj)

In the figure , E is the midpoint of AB line and CD line , and AB = CD. Write a... (answered by Alan3354)