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

2. An equilateral triangle where each side is 6 cm. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the radius of the circumcircle. [Only drawing symbols required]

3. An isosceles triangle where the base is 7.8 cm and each of the equal sides is 6.5 cm. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

4. Draw an equilateral triangle with side length 7 cm. Then draw both the circumcircle and the incircle of the triangle. Using a scale, determine the lengths of the circumradius and the inradius. Also, write whether there is any relationship between them.

5. Draw an isosceles triangle whose base is 7.8 cm and the length of each of the equal sides is 6.5 cm. Then draw an incircle (an inscribed circle) inside that triangle.

6. What is the radius of the circumcircle of a triangle whose sides are 20 cm, 21 cm, and 29 cm?

(a) 14\(\frac{1}{2}\) cm (b) 14 cm (c) 10 cm (d) 11\(\frac{1}{2}\) cm

7. Draw a right-angled triangle whose hypotenuse is 10 cm and one of the other sides is 6.5 cm. Then, draw the incircle of this triangle. (Only construction marks are required.)

8. 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.

9. 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

10. Draw a right-angled triangle whose hypotenuse is 9 cm and one of the other sides is 5.5 cm. Then draw an incircle of the triangle. (Only the construction marks are required.)

11. Draw an isosceles triangle with a base length of 5.2 cm and each of the equal sides measuring 7 cm. Then, construct the circumcircle of the triangle and measure the circumradius. (Only mark the construction steps).

12. Draw an isosceles triangle where the base length is 5 cm and each of the equal sides is 6 cm. Then, draw the incircle of the triangle. (Only construction marks should be indicated in each case.)

13. Here is the drawing of an isosceles triangle with a base length of 5.6 cm and equal sides of 7.5 cm each, along with its incircle.

14. The lengths of two sides are 7.6 cm and 6 cm, and the included angle between them is 75°. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

15. The length of one side is 6.2 cm, and the measures of the two angles adjacent to that side are 50° and 75°. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

16. A right-angled triangle in which the hypotenuse is 9 cm and one of the other sides is 5.5 cm. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

17. Draw an equilateral triangle ABC with each side measuring 5 cm. Then, draw the circumcircle of that triangle. At point A on the circle, draw a tangent. On the tangent, take a point P such that AP = 5 cm. From point P, draw another tangent to the circle, and write down which point on the circle this second tangent touches.

18. Draw a square with the same area as an isosceles triangle whose base length is 7 cm and each of the equal sides is 5 cm.

19. Draw an isosceles triangle in which each of the equal sides is 7 cm and the base is 8 cm long. Then, draw an incircle of that triangle.

20. Draw a right-angled isosceles triangle where each of the equal sides is 6 cm long. Then, draw the incircle of the triangle.

21. Draw a triangle whose two sides are 7 cm and 6 cm, and the included angle is 75°. Then draw the incircle of the triangle.

22. What is the height of an equilateral triangle ABC if each side is of length 2a units?

23. Draw an isosceles triangle with a base of 6.5 cm and two equal sides of 7 cm each. Then, draw the incircle of that triangle.

24. If the circumradius of an equilateral triangle is 4 cm, what is its inradius?

(a) 2 cm (b) 4 cm (c) 1 cm (d) 3 cm

25. If the area of the incircle of an equilateral triangle is 49 ???? cm 2 , what is the perimeter of the triangle?

(a) 21\(\sqrt3\) (b) 7\(\sqrt3\) (c) 14\(\sqrt3\) (d) 42\(\sqrt3\)

26. Draw a triangle in which two sides are 9 cm and 7 cm, and the included angle between them is 60°. Then draw the incircle of that triangle. (Only construction marks are required.)

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

28. 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.

29. 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

30. In a circle with a radius of 5 cm, AB and AC are two equal chords. The center of the circle is located outside the triangle ABC. If AB = AC = 6 cm, determine the length of the chord BC.