1. The circular measure of each interior angle of a regular hexagon is –

(a) \(\cfrac{π}{3}\) (b) \(\cfrac{2π}{3}\) (c) \(\cfrac{π}{6}\) (d) \(\cfrac{π}{4}\)

2. The circular measure of each interior angle of a regular hexagon is _____.

(a) \(\cfrac{\pi^c}{4}\) (b) \(\cfrac{\pi^c}{6}\) (c) \(\cfrac{\pi^c}{3}\) (d) \(\cfrac{2\pi^c}{3}\)

3. The circular measure of each exterior angle of a regular hexagon is –

(a) \(\cfrac{π}{3}\) (b) \(\cfrac{2π}{3}\) (c) \(\cfrac{π}{6}\) (d) \(\cfrac{π}{4}\)

4. Two circles touch each other externally at point C. AB is a common tangent to both circles and touches the circles at points A and B, respectively. Find the measure of \(\angle\)ACB.

(a) 60° (b) 45° (c) 30° (d) 90°

5. Two circles touch each other externally at point \(C\). \(AB\) is a common tangent to the two circles, touching them at points \(A\) and \(B\), respectively. The measurement of \(\angle ACB\) is –

(a) 60° (b) 45° (c) 30° (d) 90°

6. The measure of each angle of a cyclic parallelogram.

(a) 90\(^o\), 90\(^o\), 100\(^o\), 80\(^o\) (b) 100\(^o\), 80\(^o\), 100\(^o\), 80\(^o\) (c) 90\(^o\), 90\(^o\), 90\(^o\), 90\(^o\) (d) none of the above

7. What is the radius of the incircle of an equilateral triangle whose each side is \(2a\)?

(a) \(\sqrt{3a}\) cm (b) \(\cfrac{a}{\sqrt3}\) cm (c) \(\cfrac{a}{\sqrt2}\) cm (d) \(\sqrt{2a}\) cm

8. Two circles intersect at points A and B, and the circumference of each passes through the center of the other. If a straight line drawn through point A intersects the two circles again at points C and D respectively, then what type of triangle is BCD?

(a) right-angled isosceles (b) scalene (c) isosceles (d) equilateral

9. Two identical circles, each with radius \(r\), intersect in such a way that each circle passes through the center of the other. The centers of the circles are labeled A and B, and they intersect at points P and Q. The area of triangle \(\triangle APB\) will be:

(a) \(\cfrac{\sqrt3}{4}r^2\) (b) \(\cfrac{\sqrt3}{2}r^2\) (c) \(\cfrac{\sqrt3}{3}r^2\) (d) \(\sqrt3 r^2\)

10. If the sum of a central angle and an inscribed angle of a circle is 180°, then what is the measure of the central angle?

11. If both sides of a rectangle are reduced by 50%, each becomes half its original length.

(a) 100% (b) 75% (c) 25% (d) 50%

12. Prove that if a perpendicular is drawn from the right angle vertex of a right-angled triangle to the hypotenuse, then the two adjacent triangles formed are similar to each other and each is also similar to the original triangle.

13. The radius of a circle is 28 cm. What is the radian measure of the central angle subtended by an arc of length 5.5 cm on this circle?

14. If an arc of a circle measuring 220 cm in length subtends a central angle of 60°, find the radius of the circle.

15. From a ghat on one side of a 600-meter wide river, two boats set off toward the opposite bank in two different directions. If the first boat moves at an angle of 30° with the riverbank, and the second boat moves at a 90° angle to the path of the first boat, then what will be the distance between the two boats after they reach the opposite bank?

16. From a point A on the ground, the angle of elevation to the top of a vertical pillar is 30°. After moving 20 meters toward the base of the pillar and reaching point B, the angle of elevation increases to 60°. Find the height of the pillar and the distance from point A to the pillar.

17. Draw an equilateral triangle with each side measuring 7 cm. Then, draw the incircle (inscribed circle) of that triangle.

18. If a perpendicular is drawn from the right angle vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of that perpendicular are similar to each other. — Prove it.

19. If one side of a cyclic quadrilateral is extended, the exterior angle thus formed is equal to the interior opposite angle — prove it.

20. Prove that the centers of three equal circles, each touching the other, form the vertices of an equilateral triangle.

21. Draw an isosceles triangle whose base is 5.6 cm and each of the equal sides is 9 cm long. Now, construct the circumcircle of this triangle.

22. If the central angle of a circle centered at point O is 300°, then the angle subtended at the circumference by the same arc will be 60°.

23. Prove that the central angle subtended by an arc of a circle is twice any inscribed angle subtended by the same arc.

24. In drawing a histogram, the base of each rectangle for a class is taken.

(a) Statistics (b) class boundary (c) expansion (d) class width

25. If each side of an equilateral triangle is 4 cm, then the measurement of the triangle's height is —

(a) \(4\sqrt{3}\) cm (b) \(16\sqrt{3}\) cm (c) \(8\sqrt{3}\) cm (d) \(2\sqrt{3}\) cm

26. A right-angled isosceles triangle in which each of the equal sides has a length of \(a\) units. The perimeter of the triangle is —

(a) \((1+\sqrt{2})a\) units (b) \((2+\sqrt{2})a\) units (c) \(3a\) units (d) \((3+2\sqrt{2})a\) units

27. If the radius of a circle is 7 cm, then what will be the radian measure of the central angle subtended by an arc of length 5.5 cm?

28. ABCD is a cyclic trapezium in which sides AD and BC are parallel to each other. If \(\angle\)ABC = 75°, then what is the measure of \(\angle\)BCD?

(a) 105° (b) 90° (c) 150° (d) 75°

29. Draw the incircle of an isosceles triangle whose base is 5.2 cm and each of the equal sides is 7 cm long.

30. Prove that if a perpendicular is drawn from the right angle vertex of a right-angled triangle to the hypotenuse, the two triangles formed are similar to each other.