Question

D and E are midpoints of sides AB and AC of triangle ABC and O is any point on side BC. O is joined to A. If P andQ are midpoints of OB and OC respetively then DEQP is a -

Answer 1

Answer:

Step-by-step explanation:

Given: D and E are midpoints of sides AB and AC of  ΔABC.

O is any point on side BC.

P and Q are midpoints of segments OB and OC respectively.

To find: Type of Quadrilateral DEQP

Solution:

1) D and E are midpoints of sides AB and AC of  ΔABC.....given

∴By Midpoint theorem, the line joining the midpoint of any two sides of a     triangle is parallel to the third side of the triangle. The line is also half of the  third side of the triangle.

∴ seg. DE║ seg. BC

∴seg. DE= $\frac{1}{2}$ seg BC

2) Continuing from 1)

seg. DE= $\frac{1}{2}$ seg BC

seg. DE= $\frac{1}{2}$[BP+PO+OQ+QC]....P and Q are midpoints of OB and OC(given)

seg. DE= $\frac{1}{2}$[2PO+2OQ]

seg. DE= PO+OQ

seg. DE= PQ

3) In ΔABO,

DP is the segment joining the midpoints of sides AB and BO.

From midpoint theorem,

seg DP║seg AO

seg DP=$\frac{1}{2}$ seg AO

4) Similarly we can use the midpoint theorem in the ΔACO

seg EQ║seg AO

seg EQ= $\frac{1}{2}$ seg AO

5) From points 3) and 4),

seg DP║seg PO and seg EQ║seg AO

seg EQ= seg DP

Also, DE=PQ.....from 2)

Hence the quadrilateral DEQP is a parallelogram.

Answer 2

Answer:

hope this help you.............