Maharashtra Board Class 10 Maths Part II Chapter 3 Circle PDF Download

Circle

Let's Study

Circles passing through one, two, three points

Secant and tangent

Circles touching each other

Arc of a circle

Inscribed angle and intercepted arc

Cyclic quadrilateral

Secant tangent angle theorem

Theorem of intersecting chords

Let's Recall

You are familiar with the concepts regarding circle, like - centre, radius, diameter, chord, interior and exterior of a circle. Also recall the meanings of - congruent circles, concentric circles and intersecting circles.

Congruent circles have the same radius and can be in different places. Concentric circles have the same centre but different radii. Intersecting circles cross each other at two points.

Recall the properties of chord studied in previous standard and perform the activity below.

Activity I

In the adjoining figure, seg DE is a chord of a circle with centre C.

Seg CF is perpendicular to seg DE.

If diameter of the circle is 20 cm, DE = 16 cm, find CF.

To find solution of the problem, write the theorems that are useful.

(1) The perpendicular drawn from centre to a chord bisects the chord.

(2) (Write this theorem yourself)

(3) (Write this theorem yourself)

Using these properties, solve the above problem.

Teacher's Note

A perpendicular from the centre of a circle to any chord always cuts the chord into two equal parts. For example, if you draw a line from the centre of a circular plate to a straight line (chord) drawn on it, that line will cut the chord in half.

Exam Trick

Remember: If a perpendicular is drawn from the centre to a chord, it bisects (cuts in half) the chord. This means both parts are equal. Use Pythagoras theorem to find the perpendicular distance.

Points to Remember

The perpendicular from the centre of a circle to a chord bisects the chord.

Equal chords of a circle are equidistant from the centre.

Congruent chords subtend equal angles at the centre.

A diameter is the longest chord in a circle.

The perpendicular bisector of a chord passes through the centre of the circle.

Activity II

In the adjoining figure, seg QR is a chord of the circle with centre O.

P is the midpoint of the chord QR.

If QR = 24, OP = 10, find radius of the circle.

To find solution of the problem, write the theorems that are useful.

(1) (Write this theorem yourself)

(2) (Write this theorem yourself)

Using these theorems solve the problems.

Activity III

In the adjoining figure, M is the centre of the circle and seg AB is a diameter.

Seg MS is perpendicular to chord AD

Seg MT is perpendicular to chord AC

Angle DAB is congruent to angle CAB.

Prove that: chord AD is congruent to chord AC.

To solve this problem which of the following theorems will you use?

(1) The chords which are equidistant from the centre are equal in length.

(2) Congruent chords of a circle are equidistant from the centre.

Which of the following tests of congruence of triangles will be useful?

(1) SAS, (2) ASA, (3) SSS, (4) AAS, (5) hypotenuse-side test.

Using appropriate test and theorem write the proof of the above example.

Teacher's Note

Chords that are the same distance from the centre of a circle are always equal in length. Think of it like this: in a circular pizza, if you draw two chords at the same distance from the centre, both chords will have the same length.

Exam Trick

Remember: Equal distances from centre means equal chord lengths. If two chords are equidistant from the centre, they must be congruent. Write this relationship clearly in your proof.

Points to Remember

Equal chords are equidistant from the centre.

Equidistant chords from the centre are equal.

The perpendicular from the centre bisects any chord.

Use isosceles triangle properties when radii are involved.

Circles Passing Through One, Two, Three Points

In the adjoining figure, point A lies in a plane. All the three circles with centres P, Q, R pass through point A. How many more such circles may pass through point A?

If your answer is many or innumerable, it is correct.

Infinite number of circles pass through a point.

Teacher's Note

We can draw unlimited circles through a single point. Just like we can draw many circles on a paper all passing through one point marked on it, each circle has a different centre and size.

Exam Trick

Remember: One point = infinite circles. Two points = infinite circles (all centres lie on perpendicular bisector). Three non-collinear points = only one circle. Three collinear points = no circle possible.

Points to Remember

Infinite circles pass through one point.

Infinite circles pass through two distinct points.

Only one circle passes through three non-collinear points.

No circle can pass through three collinear points.

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