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. 5th Grade Pie Chart | Definition of Pie Chart | Construction |Examples

    Jul 31, 25 05:12 PM

    Pie Chart Circle
    Data can also be represented in a circle. This method, to represent data, is called a pie chart. Let us understand this method with the help of an example.

    Read More

  2. Frequency Distribution |Tally Marks |Frequency Distribution Table

    Jul 31, 25 12:23 PM

    Frequency Table
    What is frequency distribution?The number of times a particular observation occurs in a given data is called its frequency. In 7ᵗʰ grade and 8ᵗʰ grade frequency distribution,

    Read More

  3. 5th Grade Bar Graph | Definition | Interpret Bar Graphs|Free Worksheet

    Jul 31, 25 05:16 AM

    Draw a Vertical Bar Graph
    We learn how to represent the data on the bar graph. Data can be represented by bars (like rectangle) whose lengths represent numerical values. One can use horizontal or vertical bars. Instead of rect…

    Read More

  4. Construction of Bar Graphs | Examples on Construction of Column Graph

    Jul 31, 25 03:35 AM

    What is Bar Graph?
    Now we will discuss about the construction of bar graphs or column graph. In brief let us recall about, what is bar graph? Bar graph is the simplest way to represent a data. In consists of rectangular…

    Read More

  5. Successor and Predecessor | Successor of a Whole Number | Predecessor

    Jul 29, 25 12:59 AM

    Successor and Predecessor
    The number that comes just before a number is called the predecessor. So, the predecessor of a given number is 1 less than the given number. Successor of a given number is 1 more than the given number…

    Read More