Question

24. Prove that the area of the triangle formed by
joining the mid-points of the sides of a triangle
is equal to one-fourth area of the given
triangle.​

Answer 1

Answer

Given: X, Y and Z are the middle points of sides QR, RP and PQ respectively of the triangle PQR.

To prove: ar(∆XYZ) = 14 × ar(∆PQR)

Proof

1. ZY = ∥QX.

Reason:

1. Z, Y are the midpoints of PQ and PR respectively. So, using the Midpoint Theorem we get it

2. QXYZ is a parallelogram.

Reason:

2. Statement 1 implies it.

3. ar(∆XYZ) = ar(∆QZX).

Reason:

3. XZ is a diagonal of the parallelogram QXYZ.

4. ar(∆XYZ) = ar(∆RXY), and ar(∆XYZ) = ar(∆PZY).

Reason:

4. Similarly as statement 3.

5. 3 × ar(∆XYZ) = ar(∆QZX) + ar(∆RXY) = ar(∆PZY).

Reason:

5. Adding from statements 3 and 4.

6. 4 × ar(∆XYZ) = ar(∆XYZ) + ar(∆QZX) + ar(∆RXY) = ar(∆PZY).

Reason:

6. Adding ar(∆XYZ) on both side of equality in statements.

7. 4 × ar(∆XYZ) = ar(∆PQR), i.e.,

ar(∆XYZ) = 14 × ar(∆PQR). (Proved)

Reason:

7. By addition axiom for area.

Hence proved

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