If the mid-points of the sides of a quadrilateral are joined in order, prove that the area of the parallelogram so formed will be half of the area of the given quadrilateral (Fig. 9.19). [Hint: Join BD and draw perpendicular from A on BD.]

Solution:

Consider a quadrilateral ABCD

P, F, R and S are the midpoints of the sides BC, CD, AD and AB

PFRS is a parallelogram.

Join BD and BR

We know that the median of a triangle divides it into two triangles of equal areas.

BR is the median of the triangle BDA.

So, ar(BRA) = 1/2 ar(BDA) --------------- (1)

RS is the median of the triangle BRA.

So, ar(ASR) = 1/2 ar(BRA) --------------- (2)

From (1) and (2),

ar(ASR) = 1/2 (1/2 ar(BDA)

ar(ASR) = 1/4 ar(BDA) --------------------- (3)

Similarly, ar(CFP) = 1/4 ar(BCD) -------- (4)

Adding (3) and (4),

ar(ASR) + ar(CFP) = 1/4 ar(BDA) + 1/4 ar(BCD)

ar(ASR) + ar(CFP) = 1/4 ar(BCDA) --------- (5)

Similarly, ar(DRF) + ar(BSP) = 1/4 ar(BCDA) ------------ (6)

Adding (5) and (6),

ar(ASR) + ar(CFP) + ar(DRF) + ar(BSP) = 2(1/4 ar(BCDA))

ar(ASR) + ar(CFP) + ar(DRF) + ar(BSP) = 1/2 ar(BCDA) ------------- (7)

From the figure,

ar(ASR) + ar(CFP) + ar(DRF) + ar(BSP) + ar(PFRS) = ar(BCDA)

From (7),

1/2 ar(BCDA) + ar(PFRS) = ar(BCDA)

ar(PFRS) = ar(BCDA) - 1/2 ar(BCDA)

Therefore, ar(PFRS) = 1/2 ar(BCDA)

✦ Try This: In Fig, D, E and F are the mid-points of the sides BC, CA and AB respectively of △ABC. If AB = 6.2 cm, BC = 5.6 cm and CA = 4.6 cm, find the perimeter of trapezium FBCE

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 9

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NCERT Exemplar Class 9 Maths Exercise 9.3 Problem 9

If the mid-points of the sides of a quadrilateral are joined in order, prove that the area of the parallelogram so formed will be half of the area of the given quadrilateral (Fig. 9.19). [Hint: Join BD and draw perpendicular from A on BD.]

Summary:

If the mid-points of the sides of a quadrilateral are joined in order, it is proven that the area of the parallelogram so formed will be half of the area of the given quadrilateral (Fig. 9.19)

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☛ Related Questions:

• In Fig. 9.20, ABCD is a parallelogram. Points P and Q on BC trisects BC in three equal parts. Prove . . . .
• In Fig. 9.22, l, m, n, are straight lines such that l || m and n intersects l at P and m at Q. ABCD . . . .
• In Fig. 9.23, BD || CA, E is mid-point of CA and BD = 1/2 CA. Prove that ar (ABC) = 2ar (DBC)