1. From the vertex A of triangle \( \triangle ABC \), a perpendicular AD is drawn to the side BC. If \[ \frac{BD}{DA} = \frac{DA}{DC} \] then prove that triangle ABC is a right-angled triangle.

2. From vertex A of triangle \(\triangle ABC\), a perpendicular AD is drawn to the base BC, intersecting BC at point D. If \(AD^2 = BD \cdot CD\), prove that triangle ABC is a right-angled triangle and that \(\angle A = 90^\circ\).

3. From vertex \(A\) of \(\triangle ABC\), a perpendicular \(AD\) is drawn to side \(BC\). If \(\frac{BD}{DA} = \frac{DA}{DC}\), then prove that \(\triangle ABC\) is a right-angled triangle.

4. In \( \triangle ABC \), the vertex \( A \) has a perpendicular \( AD \) drawn to the side \( BC \). If \[ \frac{BD}{DA} = \frac{DA}{DC} \] then prove that \( \triangle ABC \) is a right-angled triangle.

5. In right-angled triangle ABC, \(\angle A\) is the right angle. A perpendicular AD is drawn from point A to the hypotenuse BC. Prove that: \[ \frac{∆ABC}{∆ACD} = \frac{BC^2}{AC^2} \]

6. In right-angled triangle \( \triangle ABC \), angle \( \angle A = 90^\circ \), and AO is perpendicular to BC at point O. Prove that: \[ AO^2 = BO \times CO \]

7. 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.

8. In right-angled triangle \( \triangle ABC \), \( \angle ABC = 90^\circ \) and BD is perpendicular to AC. If BD = 8 cm and AD = 4 cm, what is the length of CD?

9. 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

10. Let the smaller pillar be AB \(= x\) meters and the larger pillar be CD \(= 2x\) meters.
The midpoint of the base connection BD is O; the angles of elevation from O to the tops of the pillars are \(\angle\)AOB = \(\theta\) and \(\angle\)COD = 90\(^o-\theta\).
Since O is the midpoint of BD, we have BO = OD = \(\frac{120}{2}\) meters = 60 meters.

From \( \triangle \)ABO, we get:
\(\cfrac{AB}{BO} = \tan \theta\)
Or, \(\cfrac{x}{60} = \tan\theta\) --------(i)

From \( \triangle \)COD, we get:
\(\cfrac{CD}{OD} = \tan(90^o - \theta)\)
Or, \(\cfrac{2x}{60} = \cot\theta\) --------(ii)

Multiplying equations (i) and (ii), we get:
\(\cfrac{x}{60} \times \cfrac{2x}{60} = \tan\theta \times \cot\theta\)
Or, \(\cfrac{2x^2}{60 \times 60} = 1\)
Or, \(x^2 = \cfrac{60 \times \cancel{60}30}{\cancel{2}}\)
Or, \(x = 30\sqrt2\)

\(\therefore\) The height of the smaller pillar is \(30\sqrt2\) meters.
And the height of the larger pillar is \(30\sqrt2 \times 2\) meters \(= 60\sqrt2\) meters.
(Proved).

11. In the right-angled triangle \(ABC\), where \(\angle B = 90^\circ\), point \(D\) lies on \(AB\) such that the ratio \(AB : BC : BD = \sqrt{3} : 1 : 1\). Find the measure of \(\angle ACD\).

12. 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

13. In triangle \( \triangle ABC \), if \( \angle B \) is a right angle and \( BC = \sqrt{3} \times AB \), then what is the value of \( \sin C \)?

(a) \(\frac{1}{2}\) (b) \(\frac{1}{\sqrt2}\) (c) \(\frac{\sqrt3}{2}\) (d) 1

14. In a circle, triangle ABC is inscribed. The angle bisectors of \(\angle BAC\), \(\angle ABC\), and \(\angle ACB\) meet the circle at points X, Y, and Z respectively. Prove that in triangle ΔXYZ, \(\angle YXZ = 90^\circ - \frac{\angle BAC}{2}\).

15. A straight line intersects sides AB and AC of triangle ABC at points D and E respectively such that \(\frac{AD}{DB} = \frac{AE}{EC}\). If \(\angle ADE = \angle ACB\), prove that triangle \( \triangle ABC \) is isosceles.

16. 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 \(EF\)?

17. 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

18. In \( \triangle PQR \), ST is parallel to QR and intersects PQ at point S and PR at point T. Given that \( PQ = (X+1) \) units, \( PS = 3 \) units, \( PR = (X+6) \) units, and \( PT = 6 \) units, determine the value of \( x \).

19. A straight line intersects \( \triangle PQR \) at points \( X \) and \( Y \) on \( PQ \) and \( PR \), respectively, such that \( \frac{PX}{XQ} = \frac{PY}{YR} \). If \( \angle PXY = \angle PRQ \), then prove that \( \triangle PQR \) is an isosceles triangle.

20. In \(\triangle ABC\), the center of the incircle is \(O\), and the incircle touches the sides \(AB\), \(BC\), and \(CA\) at points \(P\), \(Q\), and \(R\) respectively. Given that \(AP = 4\) cm, \(BP = 6\) cm, \(AC = 12\) cm, and \(BC = x\) cm, determine the value of \(x\).

21. A circle is drawn with diameter \(AB\) and center \(O\). From a point \(P\) on the circle, a perpendicular is drawn to the diameter \(AB\), intersecting it at point \(N\). Prove that: \[ PB^2 = AB \cdot BN \]

22. In triangle \( \triangle ABC \), AC = BC and side BC is extended up to point D. If \( \angle ACD = 144^\circ \), then find the radian measure of each angle of triangle ABC.

23. In a right-angled triangle, if the ratio of the perpendicular (opposite side) to the hypotenuse with respect to a positive acute angle \(\theta\) is \(12 : 13\), then determine the ratio of the perpendicular to the base and the ratio of the hypotenuse to the base, and verify that \( \sec^2\theta = 1 + \tan^2\theta \).