Circles — Important Questions
Gujarat Board · Class 9 · Mathematics
45 important questions from Circles for Gujarat Board Class 9 Mathematics, with answers. Includes multiple choice and multiple correct questions.
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Important Questions from Circles
Which of the following statements about cyclic quadrilaterals are correct?
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Opposite angles sum to 180°, All four vertices lie on a circle, It can be inscribed in a circle
A cyclic quadrilateral has all four vertices on a circle, which means it can be inscribed in a circle. The key property is that opposite angles sum to 180° (Theorem 9.10). Adjacent angles are not necessarily equal, and diagonals are not necessarily equal unless it's a special case like rectangle.
An angle inscribed in a semicircle is always:
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90°
An angle inscribed in a semicircle is always a right angle (90°). This follows from Theorem 9.7: the arc of semicircle subtends 180° at center, so angle at circumference = 180° ÷ 2 = 90°. This is a fundamental property of circles.
In a circle with center O, chord AB = 6 cm and the perpendicular distance from O to AB is 4 cm. Calculate the radius of the circle.
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5 cm
The perpendicular from center bisects the chord, so half of AB = 3 cm. In the right triangle formed by radius, perpendicular distance, and half-chord: radius² = 4² + 3² = 16 + 9 = 25. Therefore, radius = √25 = 5 cm.
If ∠ABC = 70° where A, B, C are points on a circle and A, C are on opposite sides of chord BC, then ∠ADC (where D is any other point on the same side of BC as A) equals:
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70°
According to Theorem 9.8, angles in the same segment of a circle are equal. Since both ∠ABC and ∠ADC are inscribed angles subtending the same arc BC from points on the same side, they must be equal. Therefore, ∠ADC = ∠ABC = 70°.
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