1. In triangle ABC, ∠A = 90°, AB = 12 cm, AC = 5 cm, and BC = 13 cm. A perpendicular AD is drawn from point A to side BC. What is the length of AD?

2. In triangle △ABC, if ∠ABC = 90°, AB = 5 cm, and BC = 12 cm, then what is the radius of its circumcircle?

3. In triangle ABC, ∠B = 90° and BD ⊥ AC; If AB = 30 cm, BD = 24 cm, and AD = 18 cm, then what is the length of BC?

4. ABC and POR are two similar triangles. If BC = 5 cm, QR = 4 cm, and the height AD = 3 cm, then what is the length of the height PE?

(a) 4.2 cm (b) 1.25 cm (c) 5.4 cm (d) 2.4 cm

5. ABC is a right-angled triangle with hypotenuse BC. From point A, a perpendicular AD is drawn to BC. If BD = 4 cm and DC = 5 cm, then what is the length of AB?

6. In triangle ABC, ∠ABC = 90° and BD ⊥ AC; if BD = 8 cm and AD = 5 cm, then the length of CD will be –

(a) \(\cfrac{16}{5}\) cm (b) \(\cfrac{32}{5}\) cm (c) \(\cfrac{64}{5}\) cm (d) \(\cfrac{128}{5}\) cm

7. 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?

8. In triangle \( \triangle ABC \), a line parallel to side BC intersects sides AB and AC at points P and Q respectively. If \( AB = 3 \times PB \) and \( BC = 18 \) cm, then what is the length of \( PQ \)?

(a) 10 cm (b) 9 cm (c) 12 cm (d) 8 cm

9. Here’s the English translation of your math problem: In triangle \( \triangle ABC \), \( \angle ABC = 90^\circ \), \( BC = 24 \) cm, and \( E \) is the midpoint of \( AC \). If \( ED \perp BC \), then what is the length of \( BD \)?

(a) 6 cm (b) 8 cm (c) 9 cm (d) None of the above

10. ABCD is a rectangle. O is the point where the diagonals intersect. If AB = 4 cm and OD = 2.5 cm, then what is the length of BC?

(a) 4 cm (b) 1.5 cm (c) 3 cm (d) 2 cm

11. 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?

12. In triangle \(\triangle ABC\), AB = AC. Points E and F are the midpoints of sides AB and AC respectively. AD is perpendicular to BC, and AD = 4 cm. If EF = 3 cm, then what is the length of BD?

(a) 4 cm (b) 3 cm (c) 6 cm (d) 7 cm

13. AB and AC are two tangents drawn from point A to a circle with center O. The line OA intersects the chord BC (which joins the points of contact) at point M. If AM = 8 cm and BC = 12 cm, then what is the length of each tangent?

(a) 8 cm (b) 10 cm (c) 12 cm (d) 16 cm

14. A wooden box with a lid is made using wood that is 0.5 cm thick. The external dimensions of the box are: length = 20 cm, width = 16 cm, and height = 12 cm. What is the volume of the wood used to make the box?

(a) 800 cubic centimeters (b) 790 cubic centimeters (c) 820 cubic centimeters (d) 850 cubic centimeters

15. In the figure, triangle ABC is inscribed in a circle and touches the circle at points P, Q, and R. If AP = 4 cm, BP = 6 cm, AC = 12 cm, and BC = x cm, then what is the value of x?

16. In \(\triangle\)ABC, a line parallel to side BC intersects AB and AC at points P and Q, respectively. If AP = 4 cm, QC = 9 cm, and PB = AQ, then what is the length of PB?

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 isosceles triangle ABC, the circumcenter is O and \(\angle ABC = 120^\circ\). If the radius of the circle is 5 cm, find the length of side AB.

19. In the rectangular figure ABCD, diagonals AC and BD intersect at point O. If AB = 12 cm and AO = 6.5 cm, then the length of BC is __________ cm.

20. 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.

21. 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 \( \triangle ABC \)?

22. Triangle ABC is inscribed in a circle, and the circle touches the sides at points P, Q, and R respectively. If AP = 4 cm, BP = 6 cm, and AC = 12 cm, then find the length of BC.

23. In triangle ABC, \(\angle\)A = 90° and AD ⊥ BC, AC = 15 cm, AB = 20 cm, BC = 25 cm. Find the length of AD.

24. In triangle ABC, ∠ABC = 90° and BD ⊥ AC; If BD = 16 cm and AD = 10 cm, then find the length of CD.

25. In triangle \(ABC\) and triangle \(DEF\), \(\angle ABC = \angle DEF\), \(\frac{AB}{DE} = \frac{BC}{EF}\), and the ratio of their areas is \(\frac{\triangle ABC}{\triangle DEF} = \frac{25}{16}\). If \(BC = 10\) cm, what is the length of \(EF\)?