Question

In triangle PQR. X,Y,Z are the midpoint of PQ, QR and RP. prove that area of triangle XYZ is one fourth of the area of triangle PQR

Answer 1

by midpoint theorem 
xz=1/2 QR
YZ=1/2 PQ
XY=1/2PR

whick means xz/qr=1/2 and similarly

as all the sides ratio are same therefore 

triangle pqr is similar to triangle xyz

now as we know ratio of area of triangle are equal to square of the sides 

area of triangle xyz/area of triangle pqr =(1/2) ²=1/4
H.P

Answer 2

Given : ΔPQR, X, Y and Z are the midpoints of PQ, QR and RP respectively.  

To  find  :  prove that area of ΔXYZ is one fourth of the area of ΔPQR

Solution:

line joining the mid-point of two sides of a triangle is equal to half the length of the third side

XZ = QR/2  

YZ = PQ/2

XY = PR/2

XZ/QR = YZ/PQ = XY/PR =  1/2

If Corresponding sides of triangles are proportional then triangles are similar

=> ΔXYZ ≈ Δ PQR

Ratio of  Area of Similar Triangle = ( Ratio of corresponding sides)²

=> Area of ΔXYZ /  Area of ΔPQR =  (1/2)²

=> Area of ΔXYZ /  Area of ΔPQR =  1/4

=>  Area of ΔXYZ  = Area of ΔPQR/4

QED

Hence proved

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