Converse of Pythagoras’ Theorem

If in a triangle the sum of the squares of two sides is equal to the square of the third side then the triangle is a right-angled triangle, the angle between the first two sides being a right angle.

Given In the ∆XYZ, XY\(^{2}\) + YZ\(^{2}\) = XZ\(^{2}\)

Converse of Pythagoras’ Theorem Proof

To prove ∠XYZ = 90°

Construction: Draw a ∆PQR in which ∠PQR = 90° and PQ = XY, QR = YZ

Proof:

In the right-angled ∆PQR, PR\(^{2}\) = PQ\(^{2}\) + QR\(^{2}\)

Therefore, PR\(^{2}\) = XY\(^{2}\) + YZ\(^{2}\) = XZ\(^{2}\)

Therefore, PR = XZ

Now, in ∆XYZ and ∆PQR, XY = PQ, YZ = QR and XZ = PR

Therefore, ∆XYZ ≅ ∆PQR (by SSS criterion of congruency)

Therefore, ∠XYZ = ∠PQR = 90° (CPCTC)


Problems on Converse of Pythagoras’ Theorem

1. If the sides of a triangle are in the ratio 13:12:5, prove that the triangle is a right-angled triangle. Also state which angle is the right angle.

Solution:

Let the triangle be PQR.

Converse of Pythagoras’ Theorem

Here the sides are PQ = 13k, QR = 12k and RP = 5k

Now, QR\(^{2}\) + RP\(^{2}\) = (12k)\(^{2}\) + (5k)\(^{2}\)

                                           = 144k\(^{2}\) + 25k\(^{2}\)

                                           = 169k\(^{2}\)

                                           = (13k)\(^{2}\)

                                           = PQ\(^{2}\)

Therefore, by converse of Pythagoras theorem, PQR is a right-angled triangle in which ∠R = 90°.





9th Grade Math

From Converse of Pythagoras’ Theorem 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. Successor and Predecessor | Successor of a Whole Number | Predecessor

    May 24, 24 06:42 PM

    Successor and Predecessor of a Whole Number
    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

  2. Counting Natural Numbers | Definition of Natural Numbers | Counting

    May 24, 24 06:23 PM

    Natural numbers are all the numbers from 1 onwards, i.e., 1, 2, 3, 4, 5, …... and are used for counting. We know since our childhood we are using numbers 1, 2, 3, 4, 5, 6, ………..

    Read More

  3. Whole Numbers | Definition of Whole Numbers | Smallest Whole Number

    May 24, 24 06:22 PM

    The whole numbers are the counting numbers including 0. We have seen that the numbers 1, 2, 3, 4, 5, 6……. etc. are natural numbers. These natural numbers along with the number zero

    Read More

  4. Math Questions Answers | Solved Math Questions and Answers | Free Math

    May 24, 24 05:37 PM

    Math Questions Answers
    In math questions answers each questions are solved with explanation. The questions are based from different topics. Care has been taken to solve the questions in such a way that students

    Read More

  5. Estimating Sum and Difference | Reasonable Estimate | Procedure | Math

    May 24, 24 05:09 PM

    Estimating Sum or Difference
    The procedure of estimating sum and difference are in the following examples. Example 1: Estimate the sum 5290 + 17986 by estimating the numbers to their nearest (i) hundreds (ii) thousands.

    Read More