What type of triangle is formed by joining the midpoints of all sides of equilateral 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

From 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. to HOME PAGE

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

Share this page: What’s this?