Class 9 Maths Ganita Manjari Chapter 5 Exercise 5.4 Solutions– I’m Up and Down, and Round and Round

The Class 9 Maths Ganita Manjari Chapter 5 Exercise 5.4 Solutions provide detailed, step-by-step explanations to help students understand the concepts and solve each question effectively. These solutions are useful for practice, revision, and exam preparation. A PDF of the solutions is also available for easy access and revision.

NCERT Solutions for Class 9 Maths Chapter 5 Exercise Set 5.4

Exercise Set 5.4 focuses on equal chords and chords equidistant from the centre.

Class 9 Maths Ganita Manjari Chapter 5 Exercise 5.4 Solutions– I’m Up and Down, and Round and Round

Circle geometry diagram with points A, B, C, E, F, G, and H marked on chords and radii, including perpendicular segments drawn inside the circle.

Question 1. Use the Baudhāyana-Pythagoras theorem to show why chords of equal length are equidistant from the centre.

Answer:
Let AB and CD be equal chords of a circle with centre O.

Drop perpendiculars OM and ON to AB and CD respectively.

Since perpendicular from centre to chord bisects the chord:

AM = AB/2
CN = CD/2

Given:

AB = CD

So:

AM = CN

Also:

OA = OC because both are radii.

In right triangles OMA and ONC:

OA² = OM² + AM²
OC² = ON² + CN²

Since OA = OC and AM = CN:

OM² = ON²

Therefore:

OM = ON

Final answer:
Equal chords are equidistant from the centre.

Question 2. If CE is perpendicular to AB, CH is perpendicular to GF, and CE = CH, show that AB = GF.

Answer:
Since CE ⟂ AB, CE bisects chord AB.

So:

AE = EB

Since CH ⟂ GF, CH bisects chord GF.

So:

GH = HF

Now consider right triangles CEA and CHG.

CA = CG because both are radii.
CE = CH given.
∠CEA = ∠CHG = 90°.

By RHS congruence:

ΔCEA ≅ ΔCHG

So:

AE = GH

Therefore:

AB = 2AE
GF = 2GH

Since AE = GH:

AB = GF

Final answer:
AB = GF.

Question 3. Solve the previous question using the Baudhāyana-Pythagoras theorem.

Answer:
Let the radius be r.

For chord AB:

AE² = r² - CE²

For chord GF:

GH² = r² - CH²

Given:

CE = CH

Therefore:

r² - CE² = r² - CH²

So:

AE² = GH²

Hence:

AE = GH

Therefore:

AB = 2AE and GF = 2GH

So:

AB = GF

Final answer:
The two chords are equal.

NCERT Solutions for Class 9 Maths Chapter 5 - Related Links