Riders Based on Pythagoras’ Theorem

Here we will solve different types of examples on establishing riders based on Pythagoras’ theorem.

1. In the quadrilateral PQRS the diagonals PR and QS intersects at a right angle. Prove that PQ2+ RS2 = PS2 + QR2.

Diagonals are Intersects at a Right Angle


Let the diagonals intersect at O, the angle of intersection being a right angle.

In the right-angle ∆POQ, PQ2 = OP2 + OQ2.

In the right-angle ∆ROS, RS2 = OR2 + OS2.

Therefore, PQ2 + RS2 = OP2 + OQ2 + OR2 + OS2 ................. (i)

In the right-angle ∆POS, PS2 = OP2 + OS2.

In the right-angle ∆QOR, QR2 = OQ2 + OR2.

Therefore, PS2 + QR2 = OP2 + OS2 + OQ2 + OR2 ................. (ii)

From (i) and (ii), PQ2+ RS2 = PS2 + QR2. (Proved).

2. In ∆XYZ, ∠Z = 90° and ZM ⊥ XY, where M is the foot of the perpendicular. Prove that \(\frac{1}{ZM^{2}}\) = \(\frac{1}{YZ^{2}}\) + \(\frac{1}{XZ^{2}}\).

Riders Based on Pythagoras’ Theorem


In ∆XYZ and ∆ZYM,

∠XZY = ∠ZMY = 90°,

∠XYZ = ∠ZYM (Common Angle)

Therefore, by AA criterion of similarity,  ∆XYZ ∼ ∆ZYM.

\(\frac{XY}{YZ}\) = \(\frac{XZ}{ZM}\)

⟹ YZ ∙ XZ = XY ∙ ZM

Therefore, ZM = \(\frac{YZ ∙ XZ}{XY}\)

Therefore, \(\frac{1}{ZM^{2}}\) = \(\frac{XY^{2}}{YZ^{2}  ∙  XZ^{2}}\) = \(\frac{XZ^{2} + YZ^{2}}{YZ^{2}  ∙  XZ^{2}}\); [By Pythagoras’ theorem)

Therefore, \(\frac{1}{ZM^{2}}\) = \(\frac{1}{YZ^{2}}\) + \(\frac{1}{XZ^{2}}\). (Proved)

3. In ∆XYZ, ∠Z is acute and XM ⊥ YZ, M being the foot of the perpendicular. Prove that 2YZ ∙ ZM = YZ2 + ZX2 - XY2.

Riders Based on Pythagoras’ Theorem Image


From the right-angled ∆XMY,

XY2 = XM2 + YM2

         = XM2 + (YZ - ZM)2

         = XM2 + YZ2 + ZM2 - 2YZ ∙ ZM (from algebra)

         = YZ2 - 2YZ ∙ ZM + (XM2 + ZM2)

         = YZ2 - 2YZ ∙ ZM + XZ2 (from right-angled ∆XMZ)

Therefore, 2YZ ∙ ZM = YZ2 + ZX2 – XY2. (Proved)

4. Let PQRS be a rectangle. O is a point inside the rectangle. Prove that OP2 + OR2 = OQ2 + OS2.

A Point Inside the Rectangle


PQRS is a rectangle for which PQ = SR = length and QR = PS = breadth.

Join OP, OQ, OR and OS.

Draw XY through O, parallel to PQ.

As ∠QPS and ∠RSP are right angles, ∆PXO, ∆SXO, ∆RYO and ∆QYO are right-angled triangles.

Therefore, by Pythagoras’ theorem,

OP2 = PX2 + OX2,

OR2 = RY2 + OY2,

OQ2 = QY2 + OY2 and

OS2 = SX2 + OX2

Therefore, OP2 + OR2 = PX2 + OX2 + RY2 + OY2 ......... (i)

                OQ2 + OS2 = QY2 + OY2 + SX2 + OX2 ......... (ii)

But in the rectangle XSRY, SX = RY = breadth

and in the rectangle PXYQ, PX = QY = breadth.

Therefore, from (i) and (ii), OP2 + OR2 = OQ2 + OS2.

9th Grade Math

From Riders Based on 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. Addition and Subtraction of Fractions | Solved Examples | Worksheet

    Jul 18, 24 03:08 PM

    Addition and subtraction of fractions are discussed here with examples. To add or subtract two or more fractions, proceed as under: (i) Convert the mixed fractions (if any.) or natural numbers

    Read More

  2. Worksheet on Simplification | Simplify Expressions | BODMAS Questions

    Jul 18, 24 01:19 AM

    In worksheet on simplification, the questions are based in order to simplify expressions involving more than one bracket by using the steps of removal of brackets. This exercise sheet

    Read More

  3. Fractions in Descending Order |Arranging Fractions an Descending Order

    Jul 18, 24 01:15 AM

    We will discuss here how to arrange the fractions in descending order. Solved examples for arranging in descending order: 1. Arrange the following fractions 5/6, 7/10, 11/20 in descending order. First…

    Read More

  4. Fractions in Ascending Order | Arranging Fractions | Worksheet |Answer

    Jul 18, 24 01:02 AM

    Comparison Fractions
    We will discuss here how to arrange the fractions in ascending order. Solved examples for arranging in ascending order: 1. Arrange the following fractions 5/6, 8/9, 2/3 in ascending order. First we fi…

    Read More

  5. Worksheet on Comparison of Like Fractions | Greater & Smaller Fraction

    Jul 18, 24 12:45 AM

    Worksheet on Comparison of Like Fractions
    In worksheet on comparison of like fractions, all grade students can practice the questions on comparison of like fractions. This exercise sheet on comparison of like fractions can be practiced

    Read More