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) = \(\frac{1}{2}\) × 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) = \(\frac{1}{2}\) × ar(parallelogram PQRS). (Proved)

11. By addition axiom of area.

Corollaries:

(i) Are of a triangle = \(\frac{1}{2}\) × base × altitude

(ii) If a triangle and a parallelogram have equal bases and are between the same parallels then ar(triangle) = \(\frac{1}{2}\) × 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

RSS

  1. Worksheet on Yesterday, Today and Tomorrow | Sequence of the Week-days

    Nov 25, 24 01:18 AM

    Practice the worksheet on yesterday, today and tomorrow, the questions are based on the sequence of the week-days, their names and order. We know, the present day is today, the day before today

    Read More

  2. Yesterday, Today and Tomorrow | Name of each Day of the Week

    Nov 25, 24 01:09 AM

    We will learn the relation between yesterday, today and tomorrow. There are seven days in a week. We know the name of each day of the week. Sunday is known as the first day of the week.

    Read More

  3. The Story about Seasons | Spring | Summer | Autumn | Winter

    Nov 25, 24 12:48 AM

    The Four Seasons
    Kids let’s enjoy the story about seasons. Here we will discuss about the four seasons and the duration. Some months are too hot and some are too cold. The period of hot months is called the hot

    Read More

  4. Months of the Year | List of 12 Months of the Year |Jan, Feb, Mar, Apr

    Nov 25, 24 12:17 AM

    Months of the Year
    There are 12 months in a year. The months are January, February, march, April, May, June, July, August, September, October, November and December. The year begins with the January month. December is t…

    Read More

  5. Time Duration |How to Calculate the Time Duration (in Hours & Minutes)

    Nov 24, 24 11:01 PM

    Time Duration Example
    Time duration tells us how long it takes for an activity to complete. We will learn how to calculate the time duration in minutes and in hours. Time Duration (in minutes) Ron and Clara play badminton…

    Read More