# Solved Examples

First try the questions on your own and then look at the explanations below.

The sides of a quadrilateral are extended to make the angles as shown below:

What is the value of x?

Apply the theorem: Sum of all exterior angles of a polygon is 360 degrees. Hence x + 90 + 75 + 115 = 360; on solving we have x = 80.

The lengths of three sides of a triangle are known. In which of the cases given below, it is impossible to construct a triangle?

1. 15 cm, 12 cm 10 cm

2. 3.6 cm, 4.3 cm, 5.7 cm

3. 17 cm, 12 cm, 6 cm

4. 2.3 cm, 4.4 cm, 6.8 cm

Apply the theorem: Sum of any two sides is greater than the third side. We see that in (4) 2.3 + 4.4 = 6.7 which is not greater than 6.8.

In the figure given below, O is the centre of the circle. If âˆ OBC = 37Â°, find angle âˆ BAC.

In the figure there are two different circles having centres O and O' and each having the equal radius of 1 cm. Find the area of the shaded portion:

Draw the triangle as shown. Then area of half of the shaded portion = area of sector â€“ area of triangle. Also note that the angle at A is 90 degrees, since the radii are tangents to the second circle. Area of sector â€“ area of triangle = Ï€(1)2 Ã— Â¼ - Â½ (1)(1) = Ï€/4 â€“ Â½. So area of the shaded portion = 2(Ï€/4 â€“ Â½) = Ï€/2 â€“ 1.

The ratio between the number of sides of two regular polygons is 1 : 2 and the ratio between their interior angles is 2 : 3. The number of sides of these polygons are respectively:

1. 4, 8

2. 5, 10

3. 6, 12

4. 8, 16

Going by options, if the number of sides are 4 and 8, then the exterior angles of the polygons are [using 360/n] 900 and 450. So the interior angles are 900 and 1350 which are in the ratio 2 : 3. So the answer is 1st option.

1. octagon

2. hexagon

3. dodecagon

4. decagon

In this sum we have to convert the angle into degrees. Since Ï€ = 180, 5Ï€/6 = 150. Exterior angle = 30. Hence number of sides = 360/n = 360/30 = 12 and the answer would be (3).