Applying Pythagoras’ Theorem

Applying Pythagoras’ theorem we will prove the problem given below.

∆PQR is right-angle at Q. M and N are the midpoints of PQ and QR respectively. Prove that PN\(^{2}\) + RM\(^{2}\) = 5MN\(^{2}\).

Applying Pythagoras’ Theorem


Given: In ∆PQR, ∠PQR = 90°.

PM = MQ and QN = NR

Therefore, PQ = 2MQ and QR = 2QN

To prove: PN\(^{2}\) + RM\(^{2}\) = 5MN\(^{2}\).




1. ∆PQN, PQ\(^{2}\) + QN\(^{2}\) = PN\(^{2}\)

⟹ (2MQ)\(^{2}\) + QN\(^{2}\) = PN\(^{2}\)

⟹ 4MQ\(^{2}\) + QN\(^{2}\) = PN\(^{2}\)

1. By Pythagoras’ theorem


2. ∆RQM, MQ\(^{2}\) + QR\(^{2}\) = RM\(^{2}\)

⟹ MQ\(^{2}\) + (2QN)\(^{2}\) = RM\(^{2}\)

⟹ MQ\(^{2}\) + 4QN\(^{2}\) = RM\(^{2}\)

2. By Pythagoras’ theorem


3. 5MQ\(^{2}\) + 5QN\(^{2}\) = PN\(^{2}\) + RM\(^{2}\)

⟹ 5(MQ\(^{2}\) + QN\(^{2}\)) = PN\(^{2}\) + RM\(^{2}\)

3. Adding statements 1 and 2.

4. 5MN\(^{2}\) = PN\(^{2}\) + RM\(^{2}\) (Proved)

4. Applying Pythagoras’ theorem in ∆QMN.

9th Grade Math

From Converse of Pythagoras’ Theorem to HOME PAGE

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.

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.

Share this page: What’s this?

Recent Articles

  1. 2nd grade math Worksheets | Free Math Worksheets | By Grade and Topic

    Dec 06, 23 01:23 AM

    2nd Grade Math Worksheet
    2nd grade math worksheets is carefully planned and thoughtfully presented on mathematics for the students.

    Read More

  2. Rupees and Paise | Paise Coins | Rupee Coins | Rupee Notes

    Dec 04, 23 02:14 PM

    Different types of Indian Coins
    Money consists of rupees and paise; we require money to purchase things. 100 paise make one rupee. List of paise and rupees in the shape of coins and notes:

    Read More

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

    Dec 04, 23 01:50 PM

    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