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


Area of a Triangle is Half that of a Parallelogram on the Same Base and between the Same Parallels

Here we will prove that the area of a triangle is half that of a parallelogram on the same base and between the same parallels.

Given: PQRS is a parallelogram and PQM is a triangle with the same base PQ, and are between the same parallel lines PQ and SR.

To prove: ar(∆PQM) = 12 × ar(Parallelogram PQRS).

Construction: Draw MN ∥ SP which cuts PQ at N.

Proof:

            Statement

             Reason

1. SM ∥ PN

1. SR ∥ PQ being opposite sides of the parallelogram PQRS.

2. SP ∥ MN

2. By construction

3. PNMS is a parallelogram

3. By definition of parallelogram because of statements 1 and 2.

4. ar(∆PNM) = ar(∆PSM)

4. PM is a diagonal of the parallelogram PNMS.

5. 2ar(∆PNM) = ar(∆PSM) + ar(∆PNM)

5. Adding the same area on both sides of equality in statement 4.

6. 2ar(∆PNM) = ar(parallelogram PNMS)

6. By addition axiom of area.

7. MN ∥ RQ

7. A line parallel to one of the two parallel lines, is also parallel to the other line.

8. MNQR is a parallelogram.

8. Similar to statement 3.

9. 2ar(∆MNQ) = ar(parallelogram MNQR)

9. Similar to statement 6.

10. 2{ar(∆PNM) + ar(∆MNQ)} =  ar(parallelogram PNMS) + ar(parallelogram MNQR)

10. Adding statements 6 and 9.

11. 2ar(∆PQM) = ar(parallelogram PQRS), that is ar(∆PQM) = 12 × ar(parallelogram PQRS). (Proved)

11. By addition axiom of area.

Corollaries:

(i) Are of a triangle = 12 × base × altitude

(ii) If a triangle and a parallelogram have equal bases and are between the same parallels then ar(triangle) = 12 × ar(parallelogram)





9th Grade Math

From Area of a Triangle is Half that of a Parallelogram on the Same Base and between the Same Parallels 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

  1. How to Divide Decimals? | Dividing Decimals by Decimals | Examples

    May 06, 25 01:23 AM

    Dividing a Decimal by a Whole Number
    Dividing Decimals by Decimals I. Dividing a Decimal by a Whole Number: II. Dividing a Decimal by another Decimal: If the dividend and divisor are both decimal numbers, we multiply both the numbers by…

    Read More

  2. Multiplying Decimal by a Whole Number | Step-by-step Explanation|Video

    May 06, 25 12:01 AM

    Multiplying decimal by a whole number is just same like multiply as usual. How to multiply a decimal by a whole number? To multiply a decimal by a whole number follow the below steps

    Read More

  3. Word Problems on Decimals | Decimal Word Problems | Decimal Home Work

    May 05, 25 01:27 AM

    Word problems on decimals are solved here step by step. The product of two numbers is 42.63. If one number is 2.1, find the other. Solution: Product of two numbers = 42.63 One number = 2.1

    Read More

  4. Worksheet on Multiplying Decimals | Product of the Two Decimal Numbers

    May 05, 25 01:05 AM

    Practice the math questions given in the worksheet on multiplying decimals. Multiply the decimals to find the product of the two decimal numbers, same like multiplying whole numbers.

    Read More

  5. Multiplication of a Decimal by 10, 100, 1000 | Multiplying decimals

    May 05, 25 12:23 AM

    Multiplication of a Decimal by 10, 100, 1000
    The working rule of multiplication of a decimal by 10, 100, 1000, etc... are: When the multiplier is 10, 100 or 1000, we move the decimal point to the right by as many places as number of zeroes after…

    Read More