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

Solution:

Given: In ∆PQR, ∠PQR = 90°.

PM = MQ and QN = NR

Therefore, PQ = 2MQ and QR = 2QN

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

Proof:

            Statement

            Reason

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

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

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

1. By Pythagoras’ theorem

Given

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

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

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

2. By Pythagoras’ theorem

Given

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




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. 2nd Grade Geometry Worksheet | Plane and Solid Shapes | Point | Line

    Dec 12, 24 09:20 AM

    Curved Line and Straight Line
    2nd grade geometry worksheet

    Read More

  2. Types of Lines |Straight Lines|Curved Lines|Horizontal Lines| Vertical

    Dec 09, 24 10:39 PM

    Types of Lines
    What are the different types of lines? There are two different kinds of lines. (i) Straight line and (ii) Curved line. There are three different types of straight lines. (i) Horizontal lines, (ii) Ver…

    Read More

  3. Points and Line Segment | Two Points in a Curved Surface | Curve Line

    Dec 09, 24 01:08 AM

    Curved Lines and Straight Line
    We will discuss here about points and line segment. We know when two lines meet we get a point. When two points on a plane surface are joined, a straight line segment is obtained.

    Read More

  4. Solid Shapes | Basic Geometric Shapes | Common Solid Figures | Plane

    Dec 08, 24 11:19 PM

    Solid Shapes
    We will discuss about basic solid shapes. We see a variety of solid objects in our surroundings. Solid objects have one or more shapes like the following. Match the objects with similar shape.

    Read More

  5. 2nd grade math Worksheets | Free Math Worksheets | By Grade and Topic

    Dec 07, 24 03:38 PM

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

    Read More