Area of the Triangle formed by Joining the Middle Points of the Sides of a Triangle is Equal to One-fourth Area of the given Triangle.

Here we will prove that the area of the triangle formed by joining the middle points of the sides of a triangle is equal to one-fourth area of the given triangle.

Solution:

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

To prove: ar(∆XYZ) = \(\frac{1}{4}\) × ar(∆PQR)

Proof:

            Statement

            Reason

1. ZY = ∥QX.

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

2. QXYZ is a parallelogram.

2. Statement 1 implies it.

3. ar(∆XYZ) =  ar(∆QZX).

3. XZ is a diagonal of the parallelogram QXYZ.

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

4. Similarly as statement 3.

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

5. Adding from statements 3 and 4.

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

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

7. 4 × ar(∆XYZ) = ar(∆PQR), i.e., 

ar(∆XYZ) = \(\frac{1}{4}\) × ar(∆PQR). (Proved)

7. By addition axiom for area.






9th Grade Math

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