arc and segment, tangent

Lines and circles are the important elementary figures in geometry. We know that a line is a path through two or more points moving in a constant direction whereas the circle is a set of all those points in a plane that is equally distant from some fixed point. Here we will discuss the important properties of a circle in detail.

• A straight line drawn from the center of a circle to bisect a chord that is not a diameter is perpendicular to the chord.

AB is a chord of a circle with center O.

D is the midpoint of AB and OD is joined then \(OD \perp AB\)

The converse of this is also true i.e, a straight line drawn from the center of a circle perpendicular to a chord bisects the chord.

• One and only one circle can be drawn through three points not lying in the same straight line.

• Equal chords of a circle are equidistant from the center.

In a circle with center O, chord AB = chord EF.  \(OH \perp EF , OD \perp AB \) then \(OH = OD\)

The converse of this is also true i.e, chords of a circle that are equidistant from the center are equal.

• If two arcs subtend equal angles at the center they are equal.
• In equal circles or in the same circle if two chords are equal, they cut off equal arcs.

Arc PMQ subtends ∠ POQ at the center, Arc ANB subtends ∠ AOB at the center and ∠ POQ = ∠ AOB then \(\stackrel\frown{PMQ}= \stackrel\frown{ANB}\)

If Chord PQ = Chord AB then \(\stackrel\frown{PMQ}= \stackrel\frown{ANB}\)

The converse of this is also true i.e, Equal arcs subtend equal angle at the center. And if two arcs are equal the chords of the arcs are also equal.

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Let us try to solve few questions based on the above theorems.

Example 1: Prove that of any two chords of a circle, the one which is greater is nearer the center.

Given : AB > CD, prove : OP < OQ
As
 \(OP \perp AB \\ OQ \perp CD\)
And OA = OC = radius of circle.
In △ OPA, OA² = AP² + OP² and in △ OQC, OC² = CQ² + OQ²
i.e., CQ² + OQ² = AP² + OP²  as AP > CQ, therefore AP² > CQ²

to make L.H.S = R.H.S, OQ² > OP² or OQ > OP

Example 2: In equal circle with centers O and Q, find measure of ∠DQE

As  \(\stackrel\frown{AB} = \stackrel\frown{DE}\) , therefore ∠AOB = ∠DQE
5y + 5 = 7y − 43 =>2y = 48 => y = 24

∠DQE = 7 × 24 − 43= 125°

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• The angle subtended at the center by an arc of a circle is double the angle which this arc subtends at any remaining part of the circumference.
  
  A circle with center O in which arc AB subtends ∠ AOB  at the center and ∠ APB at any point on the circumference then ∠ AOB = 2 ∠ APB

 

• In a circle the angle in a

1. Semi-circle is a right angle.(figure i)
2. Segments greater than a semi-circle is less than a right angle.(figure ii)
3. Segment less than a semi-circle is greater than a right angle.(figure iii)

segment ACB in figure i is a semi-circle, therefore  ∠ ACB = 90^o, the segment in figure ii is greater than a semi-circle therefore ∠ ACB < 90^o, segment ACB is figure iii is less than a semi-circle and therefore ∠ ACB > 90^o

• Angles in the same segment of a circle are equal.

AB is a segment that subtends ∠ 1, ∠ 2 and ∠ 3 at the circumference then ∠ 1 = ∠ 2 = ∠ 3

• The opposite angle of a cyclic quadrilateral are supplementary.

If the vertices of a quadrilateral lie on a circle it is called a cyclic quadrilateral.

∠ A + ∠ C = 180° and ∠ B + ∠ D = 180°

• The tangent at any point of a circle and the radius through this point are perpendicular to each other.

O is the center of the circle and AB is tangent to circle at point P then OP \(\perp\) AB.

• If two tangents are drawn to a circle from an external point then:

1. The tangents are equal in length
2. The tangents subtend equal angles at the center of the circle.
3. The tangents are equally inclined to the line joining the point and the center of the circle.
4. The angle between the tangents is supplementary to the angle that they subtend at the center.

A circle with center O. Two tangents BQ and BP are drawn from point B to the circle.

• PB = QB
• ∠ BOQ = ∠ BOP
• ∠ QBO = ∠ PBO
• ∠ POQ + ∠ PBQ = 180°

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Example 3: What fraction of the whole circle is the arc PQ in the below figure?

Join PO and PR. If ∠ PQR = 120° then ∠ POR = 240° ( The angle subtended at the center by an arc of a circle is double the angle which this arc subtends at any remaining part of the circumference.)

240° = \(\frac{2}{3}\) of 360°, therefore Major arc PR is two-third of the circle.

Example 4: OP and OQ are tangents. If OP = 4 cm. Find OQ.

As OP = 4, therefore OQ = 4 (If two tangents are drawn to a circle from an external point are equal in length)