1. Prove that an angle in a semicircle is a right angle.

2. Match the items on the left with those on the right (any two):

3. In triangle ABC, ∠A is a right angle. From point A, a line is drawn to point D, the midpoint of the hypotenuse BC. If BC = 10 cm, then the length of AD is —

(a) 5 cm (b) 6 cm (c) 7 cm (d) 8 cm

4. 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.

5. In triangle △ABC, ∠A is a right angle and BP and CQ are medians. Prove that: \[ 5BC^2 = 4(BP^2 + CQ^2) \]

6. 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.

7. In right-angled triangle \( \triangle ABC \), where \( \angle A = 90^\circ \), a perpendicular \( AD \) is drawn from point \( A \) to the hypotenuse \( BC \). Prove that: \[ \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle ACD} = \frac{BC^2}{AC^2} \]

8. 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.

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. AB and AC are tangents to a circle with center O. Prove that AO bisects the angle between the tangents at the point of contact, forming a right angle.

11. If one side of a cyclic quadrilateral is extended, the exterior angle thus formed is equal to the interior opposite angle — prove it.

12. A right-angled isosceles triangle in which each of the equal sides has a length of \(a\) units. The perimeter of the triangle is —

(a) \((1+\sqrt{2})a\) units (b) \((2+\sqrt{2})a\) units (c) \(3a\) units (d) \((3+2\sqrt{2})a\) units

13. Given: In a circle with center O, MN is any chord that is not a diameter, and AC is a diameter. To Prove: AC > MN, i.e., the diameter is the longest chord of a circle. Construction: From the center O, draw a perpendicular OD to the chord MN. Join O and M. Proof: OM > MD [∵ Triangle OMD is a right-angled triangle and OM is the hypotenuse] Or, OA > MD [∵ OA = OM, both are radii of the same circle] Or, ½AC > ½MN Or, AC > MN Therefore, the diameter is the longest chord of a circle.

14. Prove that an angle in a segment greater than a semicircle is an acute angle.

15. Prove that an angle subtended by a semicircle is a right angle.

16. 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.

17. The central angle subtended by an arc of a circle is twice the inscribed angle subtended by the same arc—prove it.

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

19. In the triangle ABC, \(\angle A\) is a right angle. If CD is the median, prove that \(BC^2 = CD^2 + 3AD^2\).

20. 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.

21. 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.

22. The central angle formed by an arc of a circle is twice any inscribed angle formed by the same arc—prove it.

23. 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.

24. In triangle ABC, ∠A is a right angle. If CD is the median, prove that \(BC^2 = CD^2 + 3AD^2\).

25. A right-angled triangle where the hypotenuse is 12 cm and one of the other sides is 5 cm. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the radius of the circumcircle. [Only drawing symbols required]

26. A right-angled triangle in which the hypotenuse is 9 cm and one of the other sides is 5.5 cm. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

27. In triangle ABC, angle B is a right angle. If a circle is drawn with AC as the diameter and it intersects AB at point D, then which of the following statements is correct— (i) AB > AD (ii) AB = AD (iii) AB < AD

28. Prove with reasoning that an angle in a segment greater than a semicircle is an acute angle.

29. 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 \(\angle A = 90^\circ\).

30. "We have drawn a triangle PQR in which ∠Q is a right angle. If S is any point on side QR, then prove that: PS² + QR² = PR² + QS²"