1. The distance between two pillars is 150 meters. One is three times taller than the other. From the midpoint of the line segment connecting the bases of the two pillars, the angles of elevation to the tops of the pillars are complementary. Find 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 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.
4. The perimeters of two similar triangles are 27 cm and 16 cm respectively. If one side of the first triangle is 9 cm, then find the length of the corresponding side of the second triangle.
5. 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.
6. If the area of the square drawn on one side of any triangle is equal to the sum of the areas of the squares drawn on the other two sides, then prove that the angle opposite to the first side is a right angle.
7. Prove that in a triangle, if the area of the square constructed on one side is equal to the sum of the areas of the squares constructed on the other two sides, then the angle opposite the longest side is a right angle.
8. Durga was standing on a railway overbridge that is 5√3 meters high. She observed the engine of a moving passenger train at a depression angle of 30° on one side of the bridge. Two seconds later, she saw the same engine at a depression angle of 60° on the other side of the bridge. Durga's position was vertically above the railway track, which is assumed to be a straight line. Find the speed of the train.
9. If one angle of a parallelogram is 67°30′, find the radian measures of the other three angles.
10. The difference between two given angles is 40°, and their sum is \(\frac{\pi}{2}\) radians. Find the radian measures of the two angles.
11. In a right-angled triangle, the hypotenuse is 6 cm longer than one of the other two sides and 12 cm longer than the other. Find the area of the triangle.
12. Prove that if the area of a square drawn on one side of any triangle is equal to the sum of the areas of squares drawn on the other two sides, then the angle opposite the first side is a right angle.
13. “The perimeters of two similar triangles are 20 cm and 16 cm respectively. If the length of one side of the first triangle is 9 cm, what is the length of the corresponding side of the second triangle.”
14. If the sum of two angles is 135° and their difference is \(\frac{\pi^c}{12}\), find their sexagesimal and circular measures.
15. “The perimeters of two similar triangles are 24 cm and 16 cm respectively. If one side of the second triangle is 6 cm, what will be the length of the corresponding side of the first triangle.”
16. 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?
17. Prove that if a perpendicular is drawn from the right-angled vertex of any right-angled triangle to the hypotenuse, then the two resulting triangles on either side of the perpendicular are similar to each other and each is also similar to the original triangle.
18. Prove that if a perpendicular is drawn from the right-angled vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of this perpendicular are similar to each other and each is similar to the original triangle.
19. If the sum of two angles is 135° and their difference is \(\cfrac{\pi}{12}\), find the values of the two angles in degrees and radians.
20. “The perimeters of two similar triangles are 20 cm and 16 cm respectively. If the length of one side of the first triangle is 9 cm, calculate the length of the corresponding side of the second triangle.”
21. If a perpendicular is drawn from the right-angled vertex of any right triangle to the hypotenuse, then the two triangles formed on either side of this perpendicular are similar to each other, and each of them is also similar to the original triangle.
22. From the roof of a 15-meter-high house situated at one end of a park, the foot and the top of a brick kiln chimney located at the other end of the park are seen at angles of depression and elevation of 30° and 60°, respectively. Find the height of the chimney and the distance between the brick kiln and the house.
23. "Chords PQ and RS of a circle intersect each other at point X inside the circle. By joining P to S and R to Q, prove that triangles ∆PXS and ∆RSQ are similar. From this, prove that: PX × XQ = RX × XS Or, when two chords of a circle intersect internally, the product of the two segments of one chord is equal to the product of the two segments of the other chord.
24. 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.
25. In triangle ABC, angle B is a right angle. The hypotenuse is \(\sqrt{15}\), and the sum of the other two sides is 4. What is the value of \((\cos A + \cos C)\)?
(a) \(\cfrac{8}{\sqrt{13}}\) (b) \(\cfrac{-8}{\sqrt{15}}\) (c) \(\cfrac{-8}{\sqrt{13}}\) (d) \(\cfrac{8}{\sqrt{15}}\)
26. If the ratio of the areas of two similar triangles is 64:49, then find the ratio of their corresponding sides.
27. y is equal to the sum of two variables—one that varies directly with x and another that varies inversely with x. When x = y, then y = –1, and when x = 3, then y = 5. Determine the relationship between x and y.
28. 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.
29. If the sum of two angles is 135° and their difference is \(\frac{π}{12}\), write the angles in sexagesimal and circular measure.
30. y is equal to the sum of two variables—one is directly proportional to x, and the other is inversely proportional to x. When x = 1, y = -1; and when x = 3, y = 5. Determine the relationship between x and y.
