1. Draw a right-angled triangle whose hypotenuse is 9 cm and one of the other sides is 5.5 cm. Then draw an incircle of the triangle. (Only the construction marks are required.)
2. Draw a right-angled triangle whose two sides adjacent to the right angle are 4.5 cm and 6 cm. Then draw the incircle of that triangle. (Only construction marks are required.)
3. Draw a right-angled triangle whose two sides adjacent to the right angle are 7 cm and 9 cm respectively. Then draw the incircle of that triangle. (Only construction marks are required.)
4. 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).
5. Draw a triangle in which two sides are 9 cm and 7 cm, and the included angle between them is 60°. Then draw the incircle of that triangle. (Only construction marks are required.)
6. 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.
7. Draw a right-angled triangle whose two sides adjacent to the right angle are 8 cm and 6 cm respectively, and draw an incircle of the triangle. (Only construction marks are required)
8. 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]
9. Draw an isosceles triangle where the base length is 5 cm and each of the equal sides is 6 cm. Then, draw the incircle of the triangle. (Only construction marks should be indicated in each case.)
10. 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.
11. 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.
12. The three sides of a triangle are 6 cm, 8 cm, and 10 cm respectively. What is the circumradius of the triangle? This triangle is special—it’s a right triangle (since \(6^2 + 8^2 = 36 + 64 = 100 = 10^2\)). And for right triangles, the circumradius is half the hypotenuse. So the circumradius = \(\frac{10}{2} = 5\) cm.
13. Draw an isosceles triangle whose base is 7.8 cm and the length of each of the equal sides is 6.5 cm. Then draw an incircle (an inscribed circle) inside that triangle.
14. 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.
15. In a right-angled triangle, the hypotenuse is 15 cm, and the difference between the other two sides is 3 cm. Find the lengths of those two sides.
16. 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.
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. Draw a right-angled triangle with the two sides adjacent to the right angle measuring 4 cm each. Then draw the circumcircle of the triangle. (Only construction marks are required.)
19. Sathi has drawn a right-angled triangle where the length of the hypotenuse is 6 cm more than twice the length of the smallest side. If the length of the third side is 2 cm less than the hypotenuse, then calculate and write the lengths of all three sides of the triangle.
20. 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.
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. Draw a right-angled triangle whose two sides adjacent to the right angle are 4 cm and 5 cm. Then draw a circumcircle of that triangle.
23. Draw a triangle whose two sides are 7 cm and 6 cm, and the included angle is 75°. Then draw the incircle of the triangle.
