Maharashtra Board Class 9 Maths Part II Chapter 6 Circle PDF Download

Circle

Let's study.

Circle

Property of chord of the circle

Incircle

Circumcircle

Let's recall.

In adjoining figure, observe the circle with center P. With reference to this figure, complete the following table.

Let's learn.

Circle

Let us describe this circle in terms of a set of points.

The set of points in a plane which are equidistant from a fixed point in the plane is called a circle.

Some terms related with a circle.

The fixed point is called the centre of the circle.

The segment joining the centre of the circle and a point on the circle is called a radius of the circle.

The distance of a point on the circle from the centre of the circle is also called the radius of the circle.

The segment joining any two points of the circle is called a chord of the circle.

A chord passing through the centre of a circle is called a diameter of the circle.

A diameter is a largest chord of the circle.

Circles in a plane

Congruent circles have the same radii.

Concentric circles have the same centre and different radii.

Circles intersecting in a point have different centres, different radii, and only one common point.

Circles intersecting in two points have different centres, different radii, and two common points.

Teacher's Note

A circle is like a round coin or a plate. The center is the middle point. The radius is the distance from the middle to the edge.

Exam Trick

Remember: Diameter is the longest chord. It passes through the center. Two radii together make one diameter.

Points to Remember

A circle is made of all points at equal distance from one center point.
The radius is the distance from center to the circle edge.
A chord is a line joining any two points on the circle.
The diameter is a chord that passes through the center.
Congruent circles have equal radii.

Let's learn.

Properties of chord

Activity I: Every student in the group will do this activity. Draw a circle in your notebook. Draw any chord of that circle. Draw perpendicular to the chord through the centre of the circle. Measure the lengths of the two parts of the chord. Group leader will prepare a table and other students will write their observations in it.

Write the property which you have observed.

Let us write the proof of this property.

Theorem

A perpendicular drawn from the centre of a circle on its chord bisects the chord.

Given

seg AB is a chord of a circle with centre O.

seg OP \(\perp\) chord AB

To prove

seg AP \(\cong\) seg BP

Proof

Draw seg OA and seg OB

In \(\triangle OPA\) and \(\triangle OPB\)

\(\angle OPA \cong \angle OPB\) . . . . . . . . . seg OP \(\perp\) chord AB

seg OP \(\cong\) seg OP . . . . . . . . . . . . common side

hypotenuse OA \(\cong\) hypotenuse OB . . . . . . . . . . . radii of the same circle

\(\therefore \triangle OPA \cong \triangle OPB\) . . . . . . . . . hypotenuse side theorem

seg PA \(\cong\) seg PB . . . . . . . . . . . . c.s.c.t.

Activity II: Every student from the group will do this activity. Draw a circle in your notebook. Draw a chord of the circle. Join the midpoint of the chord and centre of the circle. Measure the angles made by the segment with the chord. Discuss about the measures of the angles with your friends.

Which property do the observations suggest?

Teacher's Note

When you draw a straight line from the center to a chord, it divides the chord into two equal parts. This is like cutting a pizza slice equally from the center.

Exam Trick

Remember: Perpendicular from center to chord bisects the chord. Bisect means divide into two equal parts.

Points to Remember

A perpendicular from the center divides the chord into equal parts.
The perpendicular is at 90 degrees angle to the chord.
This property helps solve many circle problems.
Equal chords are equidistant from the center.
The longer the chord, the closer it is to the center.

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