Please Help me with this midpoint problem!!!?

Question

Given: A= (x1, y1), B=(x2, y2)

M is the midpoint of Line AB.

P is the midpoint of Line BC.

Q is the midpoint of Line AC.

Prove:

A) P=( x2, y1+y2/2)

B) Q= (x1+x2/2, y1)

C) Points M and P have the same y-coordinate.

D) Points M and Q have the same x-coordinate.

E. M= (x1+x2/2, y1+y2/2)

Please explain, I'm totally lost

Thanks a bunch = )

Answer 1

use the mid point formula and solve from there, I won't give all but I'll try to start you on the right path, all other problems will be similar to this.

mid = ( x, y ) = ( (X1+X2)/2, (Y1+Y2)/2 ) gives you the x,y mid point

A) As stated above P is the mid of BC and we know P = ( X2, y1+y2/2)

therefore we know B x got added with C x and gave X2, and B's y with C's y and got y1+y2/2, since B's y is y2 we know y1 is from C.

The known, mid point P and B

B = ( x2,y2) and mid-x = x2 and midy = y1+y2/2 we know by the mid point formula.  To prove P is valid we must find C and apply the mid-point formula to get P.

C ( e,f ) Let use e  - for unkown x, of c and f for unknown y of c

we know by the mid point that

( e + x2 ) / 2 gave the midpoint result of x2 and

( f + y2 ) /2 gave the result of y1+y2/2  ( note: y1 that is from c )

doing the Cx first solve for e

e + x2 = 2x2

e = 2x2 - x2

e = x2

solve f

(f + y2)/2 = y1+y2/2

f + y2 = 2y1 + y2

f = 2y1 + y2 - y2

f = 2y1

so C( x2, 2y1 )

now we use the mid point formula for B and C and it should equal P

B ( x2,y2) C ( x2,2y1)

BC = P = ( (x2+x2)/2, (2y1 + y2 ) / 2)

solve the x portion

( X2+X2)/2 = 2X2/2 = X2

solve the y portion

(2y1+y2)/2 = y1 + y2/2

now place them back

BC ( x2, y1+y2/2 ) which equal P

so it's true P is the midpoint of BC

We had reverse engineer this problem but as you hopefully can see that's how to solve the others.  It's not a long process but there is explanations which make seem long