Triangles with Equal Areas on the Same Base have Equal Corresponding Altitudes

Here we will prove that triangles with equal areas on the same base have equal corresponding altitudes (or are between the same parallels).

Given: PQR and SQR are two triangles on the same base QR, and ar(∆PQR) = ar(∆SQC). Also, PN and SM are their corresponding altitudes.

To prove: PN = SM (or PS ∥ QR).

Construction: Join PS.




1.  \(\frac{1}{2}\) × QR × PN = \(\frac{1}{2}\) × QR × SM.

1. Are of a triangle = \(\frac{1}{2}\) × base × altitude, and ar(∆PQR) = ar(∆SQR).

2. PN = SM.

2. Cancelling \(\frac{1}{2}\) × QR from statement 1.

3. PN ∥ SM.

3. PN ⊥ QR and SM ⊥ QR.

4. PNMS is a rectangle.

4. PMNS is a parallelogram by statements 2 and 3, and two angles are right angles.

5. PN = SM (or PS ∥ QR). (Proved)

5. By statement 4, PNMS is a rectangle.

Corollary: Parallelograms with equal area on the same base have equal corresponding altitudes (or are between the same parallels).

Here, ar(parallelogram PQRS) = ar(parallelogram PQMN)

Therefore, ar(∆PRQ) = ar(∆PNQ)

Therefore, RN ∥ PQ. But RS ∥ PQ, NM ∥ PQ.

Therefore, RN ∥ RS and RN ∥ NM

Having common point (R or N), all the lines are coincident.

Therefore, the parallelogram have equal altitudes.

9th Grade Math

From Triangles with Equal Areas on the Same Base have Equal Corresponding Altitudes 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. Formation of Greatest and Smallest Numbers | Arranging the Numbers

    May 19, 24 03:36 PM

    Formation of Greatest and Smallest Numbers
    the greatest number is formed by arranging the given digits in descending order and the smallest number by arranging them in ascending order. The position of the digit at the extreme left of a number…

    Read More

  2. Formation of Numbers with the Given Digits |Making Numbers with Digits

    May 19, 24 03:19 PM

    In formation of numbers with the given digits we may say that a number is an arranged group of digits. Numbers may be formed with or without the repetition of digits.

    Read More

  3. Arranging Numbers | Ascending Order | Descending Order |Compare Digits

    May 19, 24 02:23 PM

    Arranging Numbers
    We know, while arranging numbers from the smallest number to the largest number, then the numbers are arranged in ascending order. Vice-versa while arranging numbers from the largest number to the sma…

    Read More

  4. Comparison of Numbers | Compare Numbers Rules | Examples of Comparison

    May 19, 24 01:26 PM

    Rules for Comparison of Numbers
    Rule I: We know that a number with more digits is always greater than the number with less number of digits. Rule II: When the two numbers have the same number of digits, we start comparing the digits…

    Read More

  5. Worksheets on Comparison of Numbers | Find the Greatest Number

    May 19, 24 10:42 AM

    Comparison of Two Numbers
    In worksheets on comparison of numbers students can practice the questions for fourth grade to compare numbers. This worksheet contains questions on numbers like to find the greatest number, arranging…

    Read More