Subscribe to our YouTube channel for the latest videos, updates, and tips.


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) = 14 × 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) = 14 × 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.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

RSS

  1. Worksheet on Average | Word Problem on Average | Questions on Average

    May 19, 25 02:53 PM

    Worksheet on Average
    In worksheet on average we will solve different types of questions on the concept of average, calculating the average of the given quantities and application of average in different problems.

    Read More

  2. 8 Times Table | Multiplication Table of 8 | Read Eight Times Table

    May 18, 25 04:33 PM

    Printable eight times table
    In 8 times table we will memorize the multiplication table. Printable multiplication table is also available for the homeschoolers. 8 × 0 = 0 8 × 1 = 8 8 × 2 = 16 8 × 3 = 24 8 × 4 = 32 8 × 5 = 40

    Read More

  3. How to Find the Average in Math? | What Does Average Mean? |Definition

    May 17, 25 04:04 PM

    Average 2
    Average means a number which is between the largest and the smallest number. Average can be calculated only for similar quantities and not for dissimilar quantities.

    Read More

  4. Problems Based on Average | Word Problems |Calculating Arithmetic Mean

    May 17, 25 03:47 PM

    Here we will learn to solve the three important types of word problems based on average. The questions are mainly based on average or mean, weighted average and average speed.

    Read More

  5. Rounding Decimals | How to Round a Decimal? | Rounding off Decimal

    May 16, 25 11:13 AM

    Round off to Nearest One
    Rounding decimals are frequently used in our daily life mainly for calculating the cost of the items. In mathematics rounding off decimal is a technique used to estimate or to find the approximate

    Read More