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Solved Examples

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

Example

The sides of a quadrilateral are extended to make the angles as shown below: 
            
 What is the value of x?

Solution

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.  


 

Example

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

Solution

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.


 

Example

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

Solution
Apply the theorem: Angle subtended at the center is double of the angle subtended at the circumference.  Angle at the center = 180 – 37 × 2 = 106. So ∠BAC = 106/2 = 53.  


 

Example

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:     
    

Solution

Draw the triangle as shown. Then area of half of the shaded portion = area of sector – 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 – area of triangle = π(1)2 × ¼ - ½ (1)(1) = π/4 – ½. So area of the shaded portion = 2(π/4 – ½) = π/2 – 1.  


 

Example

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

Solution

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. 


 

Example


1. octagon
2. hexagon
3. dodecagon
4. decagon

Solution

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





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