Question

Write true or false: Give reasons for your answers.
$\left( 1 \right)$ Line segment joining the center to any point on the circle is a radius of the circle.
$\left( 2 \right)$ A circle has only a finite number of equal chords.
$\left( 3 \right)$ If a circle is divided into three equal arcs, each is a major arc.
$\left( 4 \right)$ A chord of a circle, which is twice as long as its radius, is a diameter of the circle.
$\left( 5 \right)$ Sector is the region between the chord and its corresponding arc.
$\left( 6 \right)$ A circle is a plane figure.

Answer 1

Hint – In this particular question use the concept that diameter of the circle is always twice the radius of the circle, a circle has infinite equal length cords, a line joining center of the circle and any point on the circle is the radius of the circle so use these concepts to reach the solution of the question.

Complete step-by-step answer:
$\left( 1 \right)$ Line segment joining the center to any point on the circle is a radius of the circle.
As we all know that the line joining the center and any point on the circle is the radius of the circle, often denoted by r.
Consider O as the center of the circle and assume a point P on the circumference of the circle as shown in the below figure so the line joining these two points is called the radius of the circle.

So the statement (1) is true.
$\left( 2 \right)$ A circle has only a finite number of equal chords.
As we all know that the chords of the circle which are equal in length are always passing through the center of the circle, as there are infinite such cords are possible (passing through center) as shown in the figure, so the given statement is totally false.

So statement (2) is false.
$\left( 3 \right)$ If a circle is divided into three equal arcs, each is a major arc.
Consider an equilateral triangle (i.e. all sides equal) inscribed in a circle then the circle is divided into three equal arcs namely AB, BC and CA.
So these arcs are called as equal arcs (i.e. there is neither major or minor arcs only equal arcs are present) as shown in the below figure.

So statement (3) is false.
$\left( 4 \right)$ A chord of a circle, which is twice as long as its radius, is a diameter of the circle.
As we all know that the radius of the circle is a line joining the center and any point on the circle.
And we all know that diameter is twice the radius and it is also called as the longest chord of the circle, as shown in the figure.

So statement (4) is true.
$\left( 5 \right)$ Sector is the region between the chord and its corresponding arc.
As we know, a sector is a region between an arc and the two radii of the circle as shown in the figure.

So statement (5) is also false.
$\left( 6 \right)$ A circle is a plane figure.
As we know that circle is 2 – dimensional, so it is called a plane figure.
So statement (6) is true.
So this is the required answer.

Note – Whenever we face such types of questions the key concept we have to remember is that a circle is 2 – D figure so it is a plane figure, and if the circle is divided into three equal arcs then these arcs are in equal length so there are neither major arc nor minor arc present in the circle.