Question

Is it true that area of a segment of a circle is less than the area of its corresponding sector? Why?

Answer 1

Hint: Given the area of the segment of a circle and the area of the corresponding sector. We have to find that the area of the segment is less than the area of the corresponding sector. First, we will find the relation between the area of the minor segment of the circle and the minor sector of the circle and then find the relation between the area of the major segment and the major sector of the circle. Then, check the validity of the statement whether the area of a segment of a circle is less than the area of its corresponding sector.

Complete step-by-step solution:
We are given the chord. Now, we will draw the circle with chord PQ and the minor and major segment of the circle is shown.

Now, we will draw the circle showing the corresponding minor and major sectors.

Now, we will find the area of the minor segment by subtracting the area of the triangle POQ from the area of the minor sector.
$\text{ar}\text{. of minor segment}=\text{ar}\text{. of minor sector}-\text{ar}\text{. of }\mathrm{\Delta }\text{POQ}$
Here, the area of the minor segment is calculated by subtracting some value from the area of the minor sector which means the area of the minor segment is less than the area of the minor sector.
Now, we will find the relation between the area of the minor segment and its corresponding sector.
$\text{ar}\text{. of minor segment}<\text{ar}\text{. of minor sector}$
Now, we will find the area of the major segment by adding the area of the triangle POQ to the area of the major sector.
$\text{ar}\text{. of major segment}=\text{ar}\text{. of major sector}+\text{ar}\text{. of }\mathrm{\Delta }\text{POQ}$
Here, the area of the major segment is calculated by adding some value to the area of the major sector which means the area of the major segment is less than the area of the major sector.
Now, we will find the relation between the area of the major segment and its corresponding sector.
$\text{ar}\text{. of major segment}>\text{ar}\text{. of major sector}$
We can say that only the area of the minor segment is less than the area of the minor sector. But in the case of the major segment, the area is greater than its corresponding sector.
Therefore, the statement is valid only for the minor segment but not for the major segment.

Note: In such types of questions students mainly make mistakes while calculating the area of a minor segment and area of a major segment of the circle. Segments with mote area will be major segments and the segment with the low segment will be a minor segment. Also, students can make mistakes while establishing the relation between the area of a segment and its corresponding sector.