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. Properties of Multiplication | Multiplicative Identity | Whole Numbers

    Jan 15, 25 12:08 AM

    Properties of Multiplication of Whole Numbers
    There are six properties of multiplication of whole numbers that will help to solve the problems easily. The six properties of multiplication are Closure Property, Commutative Property, Zero Property…

    Read More

  2. Multiplication Table | Learn Tables from 0 – 25 | Multiplication Table

    Jan 14, 25 11:53 PM

    multiplication table
    In math multiplication table we will learn the tables from 0 – 25. These multiplication tables help the students to learn the essential multiplication facts. Multiplication tables are very important f…

    Read More

  3. 3rd Grade Math Worksheets |3rd Grade Math Sheets|3rd Grade Math Lesson

    Jan 14, 25 11:02 PM

    3rd Grade Math Worksheets
    3rd grade math worksheets is carefully planned and thoughtfully presented on mathematics for the students. Teachers and parents can also follow the worksheets to guide the students.

    Read More

  4. 3rd Grade Subtraction Worksheet | 3-Digit Subtraction Worksheets | Ans

    Jan 14, 25 01:57 PM

    Fill in the Missing Numbers Subtraction and Addition
    In 3th Grade Addition Worksheet we will solve how to subtract 3-digit numbers by expansion, subtraction of 3-digit numbers without regrouping, subtraction of 3-digit numbers with regrouping, propertie…

    Read More

  5. Facts about Subtraction | Subtraction of Small Numbers|Solved Examples

    Jan 14, 25 12:29 AM

    The operation to finding the difference between two numbers is called subtraction. Let us know some facts about subtraction which will help us to learn subtraction of large numbers. 1. Subtraction wit…

    Read More