Circles, Semi Circles and Tangents

Given: Figure 2.1 showing a circle with labelled parts.

Answer: The fixed point O is the Centre.

The centre is the fixed interior point equidistant from every point on the circumference.

Answer: The distances OA, OB, OC and OP are radii.

All line segments drawn from the centre to any point on the circumference are equal and are called radii (plural of radius).

Answer: The line segment is BC and is called the Diameter.

A diameter is the longest chord that passes through the centre of the circle, with both endpoints on the circumference.

Answer: The line touching the circle at point P is known as the Tangent.

A tangent is a straight line that touches the circle at exactly one point, called the point of tangency.

Answer: The line perpendicular to the tangent and joining the centre is named the Normal; angles ∠OPG and ∠OPF are 90°.

The normal at any point on a circle passes through the centre and is perpendicular to the tangent at that point.

Answer: One of the Chords is AC.

A chord is a line segment whose both endpoints lie on the circumference of the circle. AC connects two points on the circle without necessarily passing through the centre.

Answer: The portion of the circle with arc BP and two corresponding radii is named a Sector.

A sector is the region enclosed between two radii and the arc connecting their endpoints.

Answer: The line segment DE is known as a Chord.

DE is a chord because it joins two points on the circumference without passing through the centre.

Answer: The chord divides the circle into two parts called Segments.

The two parts formed by a chord are the minor segment (smaller part) and the major segment (larger part).

Answer: The diameter is twice the radius.

$$\text{Diameter} = 2 \times \text{Radius}$$

This is the fundamental relationship between diameter and radius of a circle.

Answer: A circle can be drawn when its Centre and Radius are given.

Knowing the centre (fixed point) and the radius (constant distance), a compass can be set to the radius and rotated about the centre to draw the circle.

Answer: A diameter divides the circle into two equal halves which are known as Semi circles.

Each half is exactly half the circle, called a semicircle, and the diameter is the boundary straight edge of each semicircle.

Answer: The point P is named the Point of Contact (also called point of tangency).

It is the unique point where the tangent line meets (touches) the circle.

Concept: When two circles touch externally, they touch at exactly one point on the outside. The distance between centres equals the sum of both radii.

$$d = R_1 + R_2$$

Answer: The distance between their centres will be the sum of their radii.

Concept: When two circles touch internally, the smaller circle lies inside the larger one and they touch at one point. The distance between centres equals the difference of the radii.

$$d = R_1 - R_2 \quad (R_1 > R_2)$$

Answer: The distance between their centres will be the difference of their radii.

Given: A circle drawn freehand (without compass).

Concept: The centre of a circle is the point of intersection of the perpendicular bisectors of any two chords.

Steps of Construction:
1. Draw a circle freehand of any convenient size.
2. Draw any two chords, say AB and CD, inside the circle.
3. Construct the perpendicular bisector of chord AB:
- With A and B as centres and radius more than half AB, draw arcs on both sides of AB. Join the intersection points to get the perpendicular bisector $l_1$.
4. Construct the perpendicular bisector of chord CD similarly to get line $l_2$.
5. The point O where $l_1$ and $l_2$ intersect is the centre of the circle.

Result: Point O is the required centre of the circle.

Given: Triangle ABC with $AB = 40$ mm, $BC = 50$ mm, $CA = 60$ mm.

Concept: The circumscribed circle (circumcircle) of a triangle passes through all three vertices. Its centre (circumcentre) is the point of intersection of the perpendicular bisectors of the sides.

Steps of Construction:
1. Draw line segment $BC = 50$ mm.
2. With B as centre and radius 40 mm, draw an arc. With C as centre and radius 60 mm, draw another arc. Their intersection gives point A.
3. Join AB and CA to complete triangle ABC.
4. Draw the perpendicular bisector of side AB.
5. Draw the perpendicular bisector of side BC.
6. Let these two perpendicular bisectors meet at point O. This is the circumcentre.
7. With O as centre and OA (= OB = OC) as radius, draw the circumcircle.

Result: The circle with centre O and radius OA passes through all three vertices A, B and C.

Given: An arc drawn freehand.

Concept: Any arc is part of a circle. The centre of that circle is equidistant from all points on the arc and can be found using perpendicular bisectors of chords of the arc.

Steps of Construction:
1. Draw a freehand arc.
2. Mark three points P, Q and R on the arc.
3. Join PQ and QR to form two chords.
4. Draw the perpendicular bisector of chord PQ.
5. Draw the perpendicular bisector of chord QR.
6. Let the two perpendicular bisectors meet at O. This is the centre of the required circle.
7. With O as centre and OP as radius, draw the complete circle.

Result: The complete circle of which the given arc is a part is obtained.