# A villager Itwaari has a plot of land of the shape of a quadrilateral. The Gram Panchayat of the village decided to take over some portion of his plot from one of the corners to construct a Health Centre. Itwaari agrees to the above proposal with the condition that he should be given equal amount of land in lieu of his land adjoining his plot so as to form a triangular plot. Explain how this proposal will be implemented.

**Solution:**

Let ABCD be the real shape of the quadrilateral field.

The proposal may be implemented as follows.

Join the BD diagonal and construct a line parallel to BD through point A.

Let that line meet further to the extended side CD of ABCD at point E.

Join BE and let BE and AD intersect each other at O.

Then, part ∆AOB is cut from the original field which is taken by the Gram Panchayat.

The new shape of the field will be ΔBCE. (See figure).

We have to prove that the area of ΔAOB is equal to the area of ΔDEO.

We can observe that ∆DEB and ∆DAB are lying on the same base BD and existing between the same parallels BD and AE.

According to Theorem 9.2: Two triangles on the same base (or equal bases) and between the same parallels are equal in area.

Therefore, ar (ΔDEB) = ar (ΔDAB)

Now, subtract ar (ΔDOB) from both side

ar (ΔDEB) - ar (ΔDOB) = ar (ΔDAB) - ar (ΔDOB)

Area (ΔDEO) = Area (ΔAOB)

Hence ΔAOB is the area of the plot taken by the gram Panchayat and an equivalent area of the plot that is ΔDEO is given to Itwaari so that the overall plot is triangular in shape that is ΔBCE.

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

**Video Solution:**

## A villager Itwaari has a plot of land of the shape of a quadrilateral. The Gram Panchayat of the village decided to take over some portion of his plot from one of the corners to construct a Health Centre. Itwaari agrees to the above proposal with the condition that he should be given equal amount of land in lieu of his land adjoining his plot so as to form a triangular plot. Explain how this proposal will be implemented.

Maths NCERT Solutions Class 9 Chapter 9 Exercise 9.3 Question 12

**Summary**

Two triangles on the same base (or equal bases) and between the same parallels are equal in area, it can be observed that areas of ΔDEO and ΔAOB are equal. Hence ΔAOB is the area of the plot taken by the gram Panchayat and an equivalent area of the plot that is ΔDEO is given to Itwaari so that the overall plot is triangular in shape that is ΔBCE.

**☛ Related Questions:**

- ABCD is a trapezium with AB || DC. A line parallel to AC intersects AB at X and BC at Y. Prove that ar (ADX) = ar (ACY).[Hint: Join CX.]
- In Fig.9.28, AP || BQ || CR. Prove that ar (AQC) = ar (PBR)
- Diagonals AC and BD of a quadrilateral ABCD intersect at O in such a way that ar (AOD) = ar (BOC). Prove that ABCD is a trapezium.
- In Fig.9.29, ar (DRC) = ar (DPC) and ar (BDP) = ar (ARC). Show that both the quadrilaterals ABCD and DCPR are trapeziums.

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