Question 31470This question is from textbook new york math a/b
: use coordinate geometry to prove that the midpoints of the sides of a kite determine a rectangle.
given: kite DEFG, DE=EF, DG=GF, K, L,M and N are midpoints of the indicated sides
Prove: KLMN is rectangle
This question is from textbook new york math a/b
Found 2 solutions by ikdeep, venugopalramana:
Answer by ikdeep(226)   (Show Source):
Answer by venugopalramana(3286)  (Show Source):
You can put this solution on YOUR website! WHAT SHAPE IS KITE ..IT IS NOT GIVEN EXCEPT THAT DE=EF AND DG=GF.HAVE YOU TYPED PROPERLY?IS IT DE=GF AND EF=DG?
I AM GIVING YOU THE METHOD BELOW..ON SOME ASSUMPTIONS..YOU CAN DO IN THE SAME WAY CHECKING THE PROBLEM PROPERLY..IF IN TROUBLE COME BACK WITH A SKETCH AND DESCRIPTION.
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ASSUMING IT A RECTANGLE AND NOT A PARALLELOGRAM...THEN....TAKE D AS ORIGIN,DE AS X AXIS AND DG AS Y AXIS..LET DE=GF=A AND EF=DG=B...THEN..COORINATES OF
D ARE (0,0)
E ARE (A,0)
F ARE (A,B)
G ARE (0,B)
SO COORDINATES OF MIDPOINTS ARE
K MIDPOINT OF DE IS (A/2,0)
L MIDPOINT OF EF IS (A,B/2)
M MIDPOINT OF FG IS (A/2,B)
N MIDPOINT OF DG IS (0,B/2)
KL=SQRT.(A^2/4 +B^2/4)
LM=SQRT.(A^2/4 +B^2/4)
MN=SQRT.(A^2/4+B^2/4)
NK=SQRT.((A^2/4+B^2/4)
ALL 4 SIDES ARE EQUAL SO IT IS A SQUARE OR RHOMBUS..
LET US FIND PRODUCT OF SLOPES OF KL AND LM...
SLOPE OF KL = (B/2)/(A/2)=B/A
SLOPE OF LM = (B/2)/(-A/2)=-B/A
PRODUCT IS NOT -1 ...SO IT IS NOT A SQUARE...SO IT IS RHOMBUS
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