Q. 1 Area of the largest triangle that can be inscribed in a semicircle of radius '𝑟' units is :

1. $r^2 Sq. units$
2. $\frac{1}{2} r^2 Sq. units$
3. $2 r^2 Sq. units$
4. $\sqrt{2}{r}^2 Sq. units$

 

(OR)

Q. 2 A cylindrical pencil sharpened at one edge is the combination of :

1. A cone and a cylinder
2. Frustum of a cone and a cylinder
3.  A hemisphere and a cylinder
4. Two cylinders

 

Answer : 1

We need to find the area of the largest triangle that can be inscribed in a semicircle of radius '𝑟' units.

In the above figure, we can say that ∠𝐶 = 90° since the angle subtended by a triangle with its base across the diameter of the semicircle is 90°

Since the radius of the semicircle is '𝑟' units, Length of the base 𝐴𝐵 = 2𝑟 𝑢𝑛𝑖𝑡𝑠

Now, the area of a triangle can be expressed as $\frac{1}{2}$ × 𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡

⇒Area of ∆𝐴𝐵𝐶 = $\frac{1}{2}$ × 𝐴𝐵 × 𝐶D

= $\frac{1}{2}$ × 2𝑟 × 𝑟 

= 𝑟² 𝑠𝑞. 𝑢𝑛𝑖𝑡s

Hence, the area of the largest triangle that can be inscribed in a semicircle of radius '𝑟' 𝑢𝑛𝑖𝑡𝑠 is 𝑟 ² 𝑠𝑞. 𝑢𝑛𝑖ts

(OR)

Answer : 1

The sharpened part of a cylindrical pencil sharpened at an edge is conical. The unsharpened part of the pencil is cylindrical.

Hence, a cylindrical pencil sharpened at an edge is a combination of a cone and a cylinder.