1. 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.
2. 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.
3. 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.
4. From the top and the base of a vertical mountain, the angles of depression and elevation to the top of a 30-meter high pillar are respectively 30° and 60°. What is the height of the mountain?
5. If the bases of two triangles lie on the same straight line and the other vertex of both triangles is common, then the ratio of their areas is ______to the ratio of the lengths of their bases.
6. 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.
7. A 5-meter-high house is located on one side of a park. From both the roof and the base of the house, the base and the top of a palm tree on the opposite side of the park are seen at angles of depression and elevation of 30° and 60°, respectively. What is the height of the palm tree? What is the distance between the house and the palm tree?
8. 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.
9. 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.
10. A 30-meter high pole and a pillar are located on the same horizontal plane. The foot of the pillar is seen from the top of the pole at a depression angle of 30°, and the top of the pillar is seen from the base of the pole at an elevation angle of 60°. What is the height of the pillar and the distance between the pole and the pillar?
11. 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).
12. From the roof of a 15-meter-high house located at one end of a park, the base and top of a brick kiln chimney at the other end of the park are seen at angles of depression and elevation of 30° and 60°, respectively. Determine the height of the brick kiln chimney and the distance between the kiln and the house.
13. The heights of two towers are 180 meters and 60 meters, respectively. From the base of the second tower, the angle of elevation to the top of the first tower is 60°. Determine the angle of elevation to the top of the second tower from the base of the first tower.
14. 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.
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. In our neighborhood, there are two houses on opposite sides of the road facing each other. If the base of a ladder is placed 6 meters away from the foot of the wall of the first house and the ladder is leaned against the wall, it makes an angle of 30° with the horizontal. But if the same ladder is placed in the same position and leaned against the wall of the second house, it makes an angle of 60° with the horizontal. (i) Find the length of the ladder. (ii) Calculate how far the foot of the ladder is from the base of the second house. (iii) Find the width of the road. (iv) Determine the height at which the top of the ladder touches the second house.
17. If two triangles have their bases on the same straight line and share the same vertex (the opposite vertex), then the ratio of their areas is equal to the ratio of the lengths of their bases.
18. 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.
19. 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
20. If a right circular cone has a base diameter of 7 cm and a slant height of 10 cm, what is the total surface area of the cone?
(a) 184.5 square cm (b) 185.4 square cm (c) 148.5 square cm (d) 184.6 square cm
21. Prove that if a perpendicular is drawn from the right angle vertex of a right-angled triangle to the hypotenuse, then the two adjacent triangles formed are similar to each other and each is also similar to the original triangle.
22. 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.
23. 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 \(h_1^2 = 3h_2^2\).
24. From a point A on the ground, the angle of elevation to the top of a vertical pillar is 30°. After moving 20 meters toward the base of the pillar and reaching point B, the angle of elevation increases to 60°. Find the height of the pillar and the distance from point A to the pillar.
25. From the roof and the base of a 16-meter high building, the angles of elevation to the top of a temple are 45° and 60°, respectively. Find the height of the temple and its horizontal distance from the building. (The base of the building and the temple lie on the same horizontal plane.)
26. A right-angled triangular prism has a base area of 100 square centimeters, and the areas of the two adjacent faces to the base are 40 square centimeters and 160 square centimeters. Its volume will be—
(a) 800 cubic cm (b) 880 cubic cm (c) 890 cubic cm (d) 900 cubic cm
27. The volume of a cone is jointly proportional to the square of the radius of the base and its height. If the ratio of the radii of two cones is 2:3 and their heights are in the ratio 5:4, then find the ratio of their volumes.
28. The lengths of the two equal sides of an isosceles triangle are 5 cm, and the base is 6 cm. The area of the triangle is —
(a) 18 square cm (b) 12 square cm (c) 15 square cm (d) 30 square cm
29. 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?
30. 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) \(30°\)
