1. The three sides of a triangle are 6 cm, 8 cm, and 10 cm respectively. What is the circumradius of the triangle? This triangle is special—it’s a right triangle (since \(6^2 + 8^2 = 36 + 64 = 100 = 10^2\)). And for right triangles, the circumradius is half the hypotenuse. So the circumradius = \(\frac{10}{2} = 5\) cm.

2. In triangle ABC, \(\angle\)BAC = 90°, and AD is perpendicular to BC. Given: AC = 8 cm, AB = 6 cm Find: The length of BD.

(a) 6 cm (b) 1.5 cm (c) 3 cm (d) 3.6 cm

3. In right-angled triangle ABC, \(\angle B = 90^\circ\), and M is the midpoint of the hypotenuse drawn from point B. If AB = 6 and AC = 8, then what is the length of BM?

(a) 10 (b) 4 (c) 16 (d) None of the above

4. In \(\triangle ABC\), \(AB = 9\) cm, \(BC = 6\) cm, and \(CA = 7.5\) cm. In \(\triangle DEF\), the corresponding side to \(BC\) is \(EF\), and \(EF = 8\) cm. Given that \(\triangle ABC \sim \triangle DEF\), determine the perimeter of \(\triangle DEF\).

5. I have drawn a right-angled triangle ABC in which the hypotenuse AB = 10 cm, the base BC = 8 cm, and the perpendicular AC = 6 cm. Find the sine and tangent values of angle ∠ABC.

6. In triangle ABC, AB = 9 cm, BC = 6 cm, and CA = 7.5 cm. In triangle DEF, side EF corresponds to side BC, and EF = 8 cm. If △DEF ~ △ABC, find the perimeter of △DEF.

(a) 22.5 cm (b) 25 cm (c) 27 cm (d) 30 cm

7. In triangle \( \triangle ABC \), if \( \angle ABC = 90^\circ \), \( AB = 6 \) cm, and \( BC = 8 \) cm, then what is the length of the circumradius of triangle \( \triangle ABC \)?

8. On triangle \( \triangle ABC \), points P and Q are such that \( \angle ABC = \angle APQ \). Given that \( AP = 3.6 \) cm, \( QC = 1.6 \) cm, and \( AQ = 4.8 \) cm, find the length of \( PB \).

(a) 1.2 cm (b) 2.4 cm (c) 6 cm (d) None of the above

9. In triangle ABC, \(\angle\)BAC = 90° and AD is perpendicular to BC. Given: AD = 8 cm, BC = 20 cm, and CD > BD Find: The length of CD.

(a) 6 cm (b) 4 cm (c) 20 cm (d) 16 cm

10. In triangle ABC, \(\angle\)ABC is a right angle, and AB = 5 cm, BC = 12 cm. What is the radius of the circumcircle of triangle ABC?

11. Draw a triangle ABC such that BC = 6 cm, CA = 5.5 cm, and AB = 4.5 cm. Then draw the incircle of ∆ABC. (Only construction marks are required.)

12. Draw triangle ABC such that BC = 7 cm, AB = 5 cm, and AC = 6 cm. Then draw the circumcircle of triangle ABC. (Only construction marks are required.)

13. Draw a triangle ABC where AB = 6 cm, BC = 10 cm, and \(\angle ABC = 45^\circ\); then draw the circumcircle of the triangle.

14. In \(\triangle\)ABC, \(\angle\)ABC = 90° and BD \(\perp\) AC. If BD = 6 cm and AD = 4 cm, then find the length of CD.

15. In a circle centered at \(O\), two chords \(AB\) and \(CD\) have equal lengths. Given that \(\angle AOB = 60°\) and the radius of the circle is \(6\) cm, find the area of \(\triangle COD\).

(a) \(9\sqrt3\) square c.m (b) \(6\sqrt3\) square c.m (c) \(2\sqrt3\) square c.m (d) \(3\sqrt3\) square c.m

16. In a circle centered at \(O\), two equal chords \(AB\) and \(CD\) are given. If \(\angle AOB = 60^\circ\) and the radius of the circle is 6 cm, determine the area of \( \triangle COD \).

(a) \(6\sqrt3\) square cm (b) \(2\sqrt3\) square cm (c) \(2\sqrt3\) square cm (d) \(9\sqrt3\) square cm

17. In \(\triangle\)ABC, \(\angle\)ABC = 90° and BD \(\bot\) AC; if AB = 6 cm, BD = 3 cm, and CD = 5.4 cm, then calculate the length of side BC.

18. In \(\triangle\)ABC, \(\angle B = 90°\) and \(BD \perp AC\). If \(AB = 6\) cm, \(BD = 3\) cm, and \(CD = 5.4\) cm, find the length of \(BC\).

19. In \(\triangle\)ABC, \(\angle\)ABC = 90°, and BD \(\bot\) AC. If BD = 6 cm and AD = 4 cm, then what is the length of CD?

20. Draw triangle ABC where AB = 8 cm, BC = 6 cm, and ∠ABC = 60°. Construct the circumcircle of that triangle. (Only construction marks are required).

21. Triangle ABC where the base BC = 5 cm, ∠ABC = 100°, and AB = 4 cm. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the length of the circumradius (i.e., the radius of the circumcircle). [Only drawing symbols required]

22. Given that \(\triangle ABC\) has \(\angle ABC = 90^\circ\) and \(BD \perp AC\); if \(BD = 6\) cm and \(AD = 4\) cm, then calculate and write the length of \(CD\).

23. In \(\triangle ABC\), \(\angle ABC = 90^\circ\) and \(BD \perp AC\); if \(AB = 6\) cm, \(BD = 3\) cm, and \(CD = 5.4\) cm, then calculate and write the length of side \(BC\).

24. In the adjacent figure, point O is located inside triangle PQR such that ∠POR = 90°, OP = 6 cm and OR = 8 cm. If PR = 24 cm and ∠QPR = 90°, then write the length of side QR.

25. In triangle ∆ABC, \(\angle ABC = 90^\circ\) and BD ⊥ AC. If BD = 8 cm and AD = 5 cm, then calculate and write the length of CD.

26. ABC is a right-angled triangle with \(\angle B\) being the right angle and BD ⟂ AC. If AD = 4 cm and CD = 16 cm, then calculate and write the lengths of BD and AB.

27. The perimeters of triangles ∆ABC and ∆DEF are 30 cm and 18 cm respectively. Given that ∆ABC ~ ∆DEF, and BC and EF are corresponding sides, if BC = 9 cm, then EF = _____________ cm.

28. Assume \(PQ\) is the tower and \(AB\) is the stick. \(∴ BC = 4\) cm, \(QC = 28\) meters Triangles \(∆PQC\) and \(∆ABC\) are similar right-angled triangles. Therefore, they are similar. \(∴ \cfrac{PQ}{AB} = \cfrac{QC}{BC} \) Or, \(\cfrac{PQ}{6} = \cfrac{28}{4}\) Or, \(PQ = \cfrac{28 \times 6}{4} = 42\) meters ∴ The height of the tower is 42 meters.

29. Draw triangle ABC in which BC = 7 cm, AB = 5 cm, and AC = 6 cm. Then construct the incircle of the triangle. (Only the construction markings are required.)

30. In right-angled triangle \(\triangle ABC\), \(\angle ABC = 90^\circ\), AB = 3 cm, BC = 4 cm. If BD is perpendicular to AC, then find the length of BD.