Let CM be perpendicular to chord AB. The triangles CMA and CMB are right triangles. Their hypotenuses, CA and CB, are equal because they are radii of the same circle, and CM is common. Hence, the triangles are congruent by the RHS Congruence Criterion. Therefore, AM = BM, proving that the perpendicuRead more
Let CM be perpendicular to chord AB. The triangles CMA and CMB are right triangles. Their hypotenuses, CA and CB, are equal because they are radii of the same circle, and CM is common. Hence, the triangles are congruent by the RHS Congruence Criterion. Therefore, AM = BM, proving that the perpendicular from the centre bisects the chord.
For more NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 5 I’m Up and Down and Round and Round (2026-27):
Since AB = AC, triangle ABC is isosceles. The altitude drawn from A to BC also bisects BC. As BC is a chord of the circumcircle, its perpendicular bisector always passes through the centre of the circle. Therefore, the altitude from A to BC passes through the centre of the circle. For more NCRead more
Since AB = AC, triangle ABC is isosceles. The altitude drawn from A to BC also bisects BC. As BC is a chord of the circumcircle, its perpendicular bisector always passes through the centre of the circle. Therefore, the altitude from A to BC passes through the centre of the circle.
For more NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 5 I’m Up and Down and Round and Round (2026-27):
Draw perpendiculars from the centre to both chords. These perpendiculars pass through the midpoints of the chords. Using the Pythagoras Theorem, the distances from the centre to the shorter and longer chords are found to be four centimetres and three centimetres, respectively. Since the chords are oRead more
Draw perpendiculars from the centre to both chords. These perpendiculars pass through the midpoints of the chords. Using the Pythagoras Theorem, the distances from the centre to the shorter and longer chords are found to be four centimetres and three centimetres, respectively. Since the chords are on opposite sides of the centre, the distance between their midpoints is seven centimetres.
For more NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 5 I’m Up and Down and Round and Round (2026-27):
We analyze the given conditional equation by moving terms to one side, which gives the simplified statement x - 2y - 6 = 0. Now, we rewrite the main polynomial expression using these exact bases, formatting it as x cube + (-2y) cube + (-6) cube minus 3 times x times (-2y) times (-6). According to alRead more
We analyze the given conditional equation by moving terms to one side, which gives the simplified statement x – 2y – 6 = 0. Now, we rewrite the main polynomial expression using these exact bases, formatting it as x cube + (-2y) cube + (-6) cube minus 3 times x times (-2y) times (-6). According to algebraic conditions, when the sum of three base terms is equal to zero, the sum of their individual cubes minus three times their product automatically equals zero. Thus, the answer is 0.
For more NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 4 Exploring Algebraic Identities (2026-27):
We rearrange the given polynomial expression into the standard conditional cubic identity template. We can rewrite the mathematical statement as x cube + y cube + (4) cube minus 3 times x times y times 4. This matches the identity where the three components are x, y and 4. The identity states that iRead more
We rearrange the given polynomial expression into the standard conditional cubic identity template. We can rewrite the mathematical statement as x cube + y cube + (4) cube minus 3 times x times y times 4. This matches the identity where the three components are x, y and 4. The identity states that if the sum of these three numbers equals zero, the total expression equals zero. Since x + y = -4, the sum x + y + 4 equals zero, making the value 0.
For more NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 4 Exploring Algebraic Identities (2026-27):
Can you explain why the converse to Theorem 4 is true, i.e., why does the perpendicular from the centre of a circle to a chord of the circle bisect the chord? (Hint: Use Fig. 5.12. You are told that ∠CMA = ∠CMB = 90°. You need to show that AM = BM.)
Let CM be perpendicular to chord AB. The triangles CMA and CMB are right triangles. Their hypotenuses, CA and CB, are equal because they are radii of the same circle, and CM is common. Hence, the triangles are congruent by the RHS Congruence Criterion. Therefore, AM = BM, proving that the perpendicuRead more
Let CM be perpendicular to chord AB. The triangles CMA and CMB are right triangles. Their hypotenuses, CA and CB, are equal because they are radii of the same circle, and CM is common. Hence, the triangles are congruent by the RHS Congruence Criterion. Therefore, AM = BM, proving that the perpendicular from the centre bisects the chord.
For more NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 5 I’m Up and Down and Round and Round (2026-27):
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-5/
See lessAn isosceles triangle ABC is inscribed in a circle, with AB = AC. Show that the altitude from A to BC passes through the centre of the circle.
Since AB = AC, triangle ABC is isosceles. The altitude drawn from A to BC also bisects BC. As BC is a chord of the circumcircle, its perpendicular bisector always passes through the centre of the circle. Therefore, the altitude from A to BC passes through the centre of the circle. For more NCRead more
Since AB = AC, triangle ABC is isosceles. The altitude drawn from A to BC also bisects BC. As BC is a chord of the circumcircle, its perpendicular bisector always passes through the centre of the circle. Therefore, the altitude from A to BC passes through the centre of the circle.
For more NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 5 I’m Up and Down and Round and Round (2026-27):
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-5/
See lessTwo parallel chords of lengths 6 cm and 8 cm are on opposite sides of the centre of a circle. If the radius of the circle is 5 cm, find the distance between the midpoints of the chords.
Draw perpendiculars from the centre to both chords. These perpendiculars pass through the midpoints of the chords. Using the Pythagoras Theorem, the distances from the centre to the shorter and longer chords are found to be four centimetres and three centimetres, respectively. Since the chords are oRead more
Draw perpendiculars from the centre to both chords. These perpendiculars pass through the midpoints of the chords. Using the Pythagoras Theorem, the distances from the centre to the shorter and longer chords are found to be four centimetres and three centimetres, respectively. Since the chords are on opposite sides of the centre, the distance between their midpoints is seven centimetres.
For more NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 5 I’m Up and Down and Round and Round (2026-27):
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-5/
See lessFind the value of x cube – 8y cube – 36xy – 216, when x = 2y + 6
We analyze the given conditional equation by moving terms to one side, which gives the simplified statement x - 2y - 6 = 0. Now, we rewrite the main polynomial expression using these exact bases, formatting it as x cube + (-2y) cube + (-6) cube minus 3 times x times (-2y) times (-6). According to alRead more
We analyze the given conditional equation by moving terms to one side, which gives the simplified statement x – 2y – 6 = 0. Now, we rewrite the main polynomial expression using these exact bases, formatting it as x cube + (-2y) cube + (-6) cube minus 3 times x times (-2y) times (-6). According to algebraic conditions, when the sum of three base terms is equal to zero, the sum of their individual cubes minus three times their product automatically equals zero. Thus, the answer is 0.
For more NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 4 Exploring Algebraic Identities (2026-27):
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-4/
See lessFind the value of x cube + y cube – 12xy + 64 when x + y = -4
We rearrange the given polynomial expression into the standard conditional cubic identity template. We can rewrite the mathematical statement as x cube + y cube + (4) cube minus 3 times x times y times 4. This matches the identity where the three components are x, y and 4. The identity states that iRead more
We rearrange the given polynomial expression into the standard conditional cubic identity template. We can rewrite the mathematical statement as x cube + y cube + (4) cube minus 3 times x times y times 4. This matches the identity where the three components are x, y and 4. The identity states that if the sum of these three numbers equals zero, the total expression equals zero. Since x + y = -4, the sum x + y + 4 equals zero, making the value 0.
For more NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 4 Exploring Algebraic Identities (2026-27):
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-4/
See less