1. Draw the mean proportional between straight line segments of lengths 12 cm and 3 cm, and determine the value of the mean proportional using a scale.
2. Draw the mean proportional between straight line segments of lengths 9 cm and 5 cm, and determine the value of the mean proportional using a scale.
3. A circle with a radius of 10 cm has two parallel chords of lengths 4 cm and 6 cm. What is the distance between the two chords?
(a) \(\sqrt7(\sqrt3-\sqrt{12})\) cm (b) \(\sqrt3(\sqrt7+\sqrt{12})\) cm (c) \(\sqrt{13}(\sqrt{12}-\sqrt{7})\) cm (d) \(\sqrt{7}(\sqrt{13}+\sqrt{12})\) cm
4. The radii of two circles are 4 cm and 3 cm respectively, and the distance between their centers is 13 cm. Find the length of a direct common tangent between the two circles.
5. If two circles with radii of 7 cm and 3 cm touch externally, then the distance between their centers will be 4 cm.
6. The radii of two circles are 4 cm and 3 cm respectively, and the distance between their centers is 13 cm. Find the length of a direct common tangent to the two circles.
7. If the lengths of two equal chords are _____ cm and the distance between them is 4 cm, then the diameter of the circle will be 10 cm.
8. In a right-angled triangle, the hypotenuse is 15 cm, and the difference between the other two sides is 3 cm. Find the lengths of those two sides.
9. Using the geometric method, construct the mean proportional between 4 cm and 3 cm. (Only construction steps are to be shown.)
10. In a right-angled triangle, the lengths of the two sides adjacent to the right angle are 4 cm and 3 cm. If the triangle is rotated once completely about the longer of these two sides as the axis, the solid formed is a cone. Calculate and write the lateral surface area, total surface area, and volume of the cone formed.
11. The radii of two circles are 5 cm and 3 cm. If the two circles are externally tangent to each other, the distance between their centers will be -
(a) 2 cm (b) 2.5 cm (c) 1.5 cm (d) 8 cm
12. AB and CD are two parallel straight lines. AD and BC intersect each other at point O. If OA = 2 cm, OB = 3 cm, and OD = 4 cm, then what is the length of OC?
(a) 6 cm (b) 4 cm (c) 4.8 cm (d) 4.2 cm
13. A right-angled triangle has two sides adjacent to the right angle measuring 4 cm and 3 cm. If the triangle is rotated once completely around its hypotenuse as the axis, find the volume of the solid formed.
14. In a right-angled triangle, the lengths of the two sides adjacent to the right angle are 4 cm and 3 cm respectively. If the triangle is rotated once completely around the longer of the two adjacent sides, find the total surface area and the volume of the solid formed.
15. Two parallel chords AB and CD of lengths 10 cm and 24 cm respectively lie on opposite sides of the center O of a circle. If the distance between the chords AB and CD is 17 cm, find the radius of the circle.
16. Two parallel chords of a circle with a radius of 5 cm have lengths of 6 cm and 8 cm. What is the distance between the two chords?
17. The radii of two circles are 3.5 cm and 2 cm. The circles touch each other internally. The distance between the centers of the two circles will be –
(a) 5.5 cm (b) 1 cm (c) 1.5 cm (d) None of the above
18. Two identical circles, each with a radius of 13 cm, intersect each other, and the length of their common chord is 10 cm. What is the distance between the centers of the two circles?
19. "The differences between the semi-perimeter of a triangle and the lengths of its sides are respectively 4 cm, 7 cm, and 5 cm. The area of the triangle is –"
(a) \(20\sqrt{7}\) square cm (b) \(10\sqrt{14}\) square cm (c) \(20\sqrt{14}\) square cm (d) 140 square cm
20. In a circle, AB and AC are two mutually perpendicular chords. If AB = 4 cm and AC = 3 cm, then the radius of the circle is _____ cm.
21. The radii of two circles are 8 cm and 3 cm, respectively, and the distance between their centers is 13 cm. Find the length of a common external tangent of the circles.
(a) 10 cm (b) 14 cm (c) 15 cm (d) 12 cm
22. Determine the mean proportional between 4 cm and 3 cm using the geometric method. (In each step, only drawing markings should be shown).
23. In a circle centered at O, there are two chords of lengths 6 cm and 8 cm. If the distance from the center to the shorter chord is 4 cm, calculate and write the distance from the center to the other chord.
24. In a circle centered at O, there are two parallel chords AB and CD with lengths 10 cm and 24 cm, positioned on opposite sides of the center. If the distance between the chords AB and CD is 17 cm, then calculate and write the radius of the circle.
25. Draw a straight line segment XY with a length of 4 cm, and use XY as the diameter to draw a circle. At points X and Y, draw tangents to the circle, and write about the relationship between these two tangents.
26. The radii of two circles are 8 cm and 3 cm respectively, and the distance between their centers is 1.3 cm. Find the length of a direct common tangent between the two circles.
27. A circle has a radius of 2.5 cm. Two chords AB = 4 cm and AC = 3 cm are drawn. What is the measure of angle ∠BAC?
28. Two chords of a circle have their lengths in the ratio 4:3. If their distances from the center are 9 cm and 12 cm respectively, what is the length of the chord closer to the center?
29. Two circles are touching each other externally, and the distance between their centers is 7 cm. If the radius of one circle is 4 cm, find the radius of the other circle.
