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. |
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