1. The distance between two pillars is 150 meters. One is three times as tall as the other. From the midpoint of the line connecting the bases of the two pillars, the angles of elevation to their tops are complementary. What is the height of the shorter pillar?

2. The distance between two pillars is 150 meters. One pillar's height is three times the other. The angles of elevation to their tops from the midpoint of the line joining their bases are complementary. Find the height of the smaller pillar.

3. Two pillars are 120 meters apart, with one pillar's height being twice that of the other. At the midpoint of the line segment connecting their bases, the angles of elevation to the tops of the pillars are complementary. Determine the heights of the pillars.

4. Two poles are placed 120 meters apart, and the height of one pole is double that of the other. If the angles of elevation to the tops of the two poles from the midpoint of the line joining their bases are complementary, find the height of the shorter pole.

5. Here is the English translation: > Two pillars of equal height are located directly opposite each other at points A and B on either side of a 120-meter wide road. From point C, on the line joining their bases, the angles of elevation to the tops of the pillars at A and B are 60° and 30°, respectively. Find the length of AC.

6. Two points on the ground lie along the same straight line with the base of a vertical pillar. From these two points, the angles of elevation to the top of the pillar are complementary. If the distances from the two points to the base of the pillar are 9 meters and 16 meters respectively, and both points are on the same side of the pillar, find the height of the pillar.

7. 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).

8. The ratio of the heights of two pillars is 1:3. If the angle of elevation from the base of the shorter pillar to the top of the taller pillar is 60°, then find the angle of elevation from the base of the taller pillar to the top of the shorter pillar.

9. If the angles of depression from the top of a lighthouse to the bases of the masts of two ships situated along the same straight line are 60° and 30° respectively, and the nearer ship is 150 meters away from the lighthouse, then calculate the distance of the farther ship from the lighthouse and the height of the lighthouse.

10. From two points located on the same side and along the same horizontal line passing through the base of a vertical pillar, the angles of elevation to the top of the pillar are respectively \(\theta\) and \(\phi\). If the height of the pillar is \(h\), find the distance between the two points.

11. The heights of two pillars are 180 meters and 60 meters respectively. If the angle of elevation from the base of the second pillar to the top of the first pillar is 60°, find the angle of elevation from the base of the first pillar to the top of the second pillar.

12. The heights of two pillars are 180 meters and 60 meters respectively. If the angle of elevation to the top of the second pillar from the base of the first pillar is 30°, then find the angle of elevation to the top of the first pillar from the base of the second pillar.

13. The heights of two pillars are 45 meters and 15 meters respectively. From the base of the second pillar, the angle of elevation to the top of the first pillar is 60°. Determine the angle of elevation to the top of the second pillar from the base of the first pillar.

14. From a lighthouse, the angles of depression to the bases of the masts of two ships located along the same straight line are 60° and 30°, respectively. If the mast of the nearer ship is 150 meters away from the lighthouse, determine how far the mast of the farther ship is from the lighthouse and also find the height of the lighthouse.

15. The heights of two pillars are \(180\) meters and \(60\) meters, respectively. The angle of elevation from the base of the second pillar to the top of the first pillar is \(60°\). Determine the angle of elevation from the base of the first pillar to the top of the second pillar.

(a) \(90°\) (b) \(60°\) (c) \(45°\) (d) \(30°\)

16. The heights of two pillars are 180 meters and 60 meters, respectively. If the angle of elevation to the top of the first pillar from the base of the second pillar is 60°, then determine the angle of elevation to the top of the second pillar from the base of the first pillar.

17. The heights of two pillars are 180 meters and 60 meters, respectively. If the angle of elevation from the base of the second pillar to the top of the first pillar is 60°, determine the angle of depression from the base of the first pillar to the top of the second pillar.

18. The heights of two pillars are 180 meters and 60 meters. If the angle of elevation from the base of the second pillar to the top of the first pillar is 60°, what is the angle of elevation from the base of the first pillar to the top of the second pillar?

19. The heights of two pillars are 180 meters and 60 meters, respectively. If the angle of elevation to the top of the first pillar from the base of the second pillar is 60°, determine the angle of elevation to the top of the second pillar from the base of the first pillar.

20. The heights of two pillars are 180 meters and 60 meters, respectively. If the angle of elevation to the top of the first pillar from the base of the second pillar is 60°, calculate and write the angle of elevation to the top of the second pillar from the base of the first pillar.

21. If the angles of depression from a lighthouse to two ships located along the same straight line are 60° and 30°, and the nearer ship is 150 meters away from the lighthouse, then what is the distance of the farther ship from the lighthouse?

22. The heights of two pillars are \(h_1\) meters and \(h_2\) meters respectively. If the angle of elevation from the base of the second pillar to the top of the first pillar is 60°, and the angle of elevation from the base of the first pillar to the top of the second pillar is 45°, show that \(h_1^2 = 3h_2^2\).

23. The heights of two pillars are 120 meters and 40 meters respectively. If the angle of elevation from the base of the second pillar to the top of the first pillar is 60°, then what is the angle of elevation from the base of the first pillar to the top of the second pillar?

24. The English translation is: "The heights of two pillars are 180 meters and 60 meters respectively. If the angle of elevation from the base of the second pillar to the top of the first is 60°, determine the angle of elevation from the base of the first pillar to the top of the second."

25. Relative to a point on a horizontal plane, a kite is observed at an elevation angle of \(45^\circ\) when it is at a height of 45 meters, and at an elevation angle of \(60^\circ\) when it is at a height of 60 meters. > Show that the distance between these two positions of the kite is > \[ 5\sqrt{6(23 - 12\sqrt{3})} \text{ meters}, \] > assuming the horizontal distance from the point to the kite remains constant.