1. 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.

2. 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.

3. Prove that the area of the square drawn on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the squares drawn on the other two sides.

4. Prove that in a right-angled triangle, the area of the square constructed on the hypotenuse is equal to the sum of the areas of the squares constructed on the other two sides.

5. Prove that in a right-angled triangle, the area of the square constructed on the hypotenuse is equal to the sum of the areas of the squares constructed on the other two sides.

6. The area of the square drawn on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the squares drawn on the other two sides.

7. 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.

8. 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.

9. If a perpendicular is drawn from the right angle vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of that perpendicular are similar to each other. — Prove it.

10. Prove that if a perpendicular is drawn from the right-angle vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of the perpendicular are similar to each other.

11. 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, and each of these triangles is similar to the original triangle.

12. 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.

13. "Prove that, in any rhombus, the sum of the squares drawn on its sides is equal to the sum of the squares drawn on its diagonals."

14. Prove that if a perpendicular is drawn from the right-angle vertex of a right triangle to the hypotenuse, the two triangles formed on either side of the perpendicular are similar to the original triangle and also similar to each other.

15. "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.

16. Two tangents are drawn to a circle from points A and B on the circumference, and they intersect at point C. Another point P lies on the circumference, on the side opposite to where point C is located with respect to the center. If \(\angle\)APB = 35°, then what is the measure of \(\angle\)ACB?

(a) 145° (b) 55° (c) 110° (d) None of the above

17. 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.

18. 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.

19. 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.

20. "Prove that, in a triangle, a straight line drawn parallel to the second side through the midpoint of one side bisects the third side. [Prove using Thales's Theorem]"

21. Amina has drawn a right-angled triangle with the two sides adjacent to the right angle measuring 15 cm and 20 cm. When the triangle is revolved once completely around the 15 cm side as the axis, it forms a solid. Calculate the lateral surface area, the total surface area, and the volume of the solid formed.

22. In a right-angled triangle, the lengths of the two sides adjacent to the right angle are 4 cm and 3 cm respectively. If the triangle is rotated once completely around the longer of the two adjacent sides, find the total surface area and the volume of the solid formed.

23. Given: In triangle △ABC, O is the circumcenter and OD ⊥ BC. Prove that: ∠BOD = ∠BAC Let’s break it down in English: **Given:** In triangle △ABC, O is the circumcenter (the point where the perpendicular bisectors of the sides meet), and OD is perpendicular to side BC. **To Prove:** The angle ∠BOD formed at the center between points B and D is equal to the angle ∠BAC at the vertex A. This is a classic geometry result based on the properties of a circle and triangle. Would you like me to walk you through the full proof in English as well?

24. From a ghat on one side of a 600-meter wide river, two boats set off toward the opposite bank in two different directions. If the first boat moves at an angle of 30° with the riverbank, and the second boat moves at a 90° angle to the path of the first boat, then what will be the distance between the two boats after they reach the opposite bank?

25. 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.

26. 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.

27. 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).

28. In a right-angled triangle, the lengths of the two sides adjacent to the right angle are 4 cm and 3 cm. If the triangle is rotated once completely about the longer of these two sides as the axis, the solid formed is a cone. Calculate and write the lateral surface area, total surface area, and volume of the cone formed.

29. Exactly — well explained. Since triangle ABC is isosceles with AC = BC, and AB² equals the sum of the squares of AC and BC, this satisfies the converse of the Pythagorean theorem. That confirms ∠C is a right angle, measuring **90 degrees**. Want to try applying this to a new triangle scenario? I’ve got a few twists if you’re up for it!

(a) 9 meters (b) 10 meters (c) 11meters (d) 12 meters

30. Two triangles \(∆PQR\) and \(∆SQR\) are drawn on the same base \(QR\) and on the same side of it, having equal areas. Let \(F\) and \(G\) be the centroids of the triangles respectively. Prove that: \(FG ∥ QR\).