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Let ABCD be a cyclic quadrilateral. Explain why the exterior angle at any vertex is equal to the interior opposite angle (e.g., ∠CDE = ∠ABC, where E is a point on the extension of side CD).
Let ABCD be a cyclic quadrilateral and let CD be extended to E. Then ∠CDE and ∠CDA form a linear pair, so their sum is 180°. Also, opposite angles of a cyclic quadrilateral are supplementary. Therefore, ∠ABC + ∠CDA = 180° and ∠CDE + ∠CDA = 180°. Hence, ∠CDE = ∠ABC. Thus, the exterior angle of a cyclRead more
Let ABCD be a cyclic quadrilateral and let CD be extended to E. Then ∠CDE and ∠CDA form a linear pair, so their sum is 180°. Also, opposite angles of a cyclic quadrilateral are supplementary.
Therefore,
∠ABC + ∠CDA = 180°
and
∠CDE + ∠CDA = 180°.
Hence,
∠CDE = ∠ABC.
Thus, the exterior angle of a cyclic quadrilateral is equal to its interior opposite angle.
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 lessA quadrilateral MNOP is inscribed in a circle. If MN is a diameter, what can you say about ∠MOP and ∠MNP? Explain your reasoning.
Since MNOP is inscribed in a circle, both ∠MOP and ∠MNP subtend the same arc MP. ∠MOP is formed at the centre of the circle, whereas ∠MNP is formed at a point on the circumference. According to the theorem, the angle subtended by an arc at the centre is twice the angle subtended by the same arc at aRead more
Since MNOP is inscribed in a circle, both ∠MOP and ∠MNP subtend the same arc MP. ∠MOP is formed at the centre of the circle, whereas ∠MNP is formed at a point on the circumference. According to the theorem, the angle subtended by an arc at the centre is twice the angle subtended by the same arc at any point on the remaining circle.
Therefore,
∠MOP = 2∠MNP.
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 lessA regular hexagon is inscribed in a circle of radius r. Find the length of the sides of the hexagon and the distance of each side from the centre of the circle.
A regular hexagon divides the circle into six equal sectors. Therefore, the central angle corresponding to each side is 60°. The triangle formed by two radii and one side has two sides equal to r and included angle 60°, so it is equilateral. Hence, each side of the hexagon is r. The perpendicular diRead more
A regular hexagon divides the circle into six equal sectors. Therefore, the central angle corresponding to each side is 60°. The triangle formed by two radii and one side has two sides equal to r and included angle 60°, so it is equilateral. Hence, each side of the hexagon is r.
The perpendicular distance from the centre to a side is the altitude of this equilateral triangle.
Therefore, distance = (√3/2)r.
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 10 cm and 24 cm are on the same side of the centre of a circle. The distance between the chords is 7 cm. Find the radius of the circle.
The half-lengths of the chords are 5 cm and 12 cm. Since the longer chord is closer to the centre, let its distance from the centre be x. Then the distance of the 10 cm chord is x + 7. Using right triangles: r² = x² + 12² r² = (x + 7)² + 5² Therefore, x² + 144 = x² + 14x + 49 + 25 14x = 70 x = 5 ThuRead more
The half-lengths of the chords are 5 cm and 12 cm. Since the longer chord is closer to the centre, let its distance from the centre be x. Then the distance of the 10 cm chord is x + 7.
Using right triangles:
r² = x² + 12²
r² = (x + 7)² + 5²
Therefore,
x² + 144 = x² + 14x + 49 + 25
14x = 70
x = 5
Thus, r² = 25 + 144 = 169, so r = 13 cm.
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 lessIn a circle with centre O, chords AB and AC are congruent. Explain why this statement is true: “The centre of the circle lies on the angle bisector of ∠BAC”.
Let AB and AC be congruent chords of a circle with centre O. Equal chords are at equal distances from the centre. Therefore, the perpendicular distances from O to AB and AC are equal. Alternatively, in triangles AOB and AOC, OA is common, OB = OC because they are radii andAB = AC because the chordsRead more
Let AB and AC be congruent chords of a circle with centre O. Equal chords are at equal distances from the centre. Therefore, the perpendicular distances from O to AB and AC are equal. Alternatively, in triangles AOB and AOC, OA is common, OB = OC because they are radii andAB = AC because the chords are congruent. Thus, the triangles are congruent. Hence, ∠BAO = ∠OAC. Therefore, AO bisects ∠BAC.
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 less