College Geometry- Circles

Dr. Layth M. Alabdulsada

Circle

Definition 1. A circle is the set of points lying at a given distance R from a given point Oin the plane.

The centre; equivalently it is the curve traced out by a point that moves in a plane sothat its distance from a given point is constant.

The distance between any point of the circle and the centre is called the radius.

The circumference is the length of one circuit along the circle, or the distance around thecircle.

Figure 1: A circle (black), which is measured by its circumference (C), diameter (D) in cyan,and radius (R) in red; its centre (O) is in magenta

Terminology

• Arc: any connected part of a circle. Specifying two end points of an arc and a centerallows for two arcs that together make up a full circle.

• Chord: a line segment whose endpoints lie on the circle, thus dividing a circle into twosegments.

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• Diameter : a line segment whose endpoints lie on the circle and that passes throughthe centre; or the length of such a line segment. This is the largest distance betweenany two points on the circle. It is a special case of a chord, namely the longest chordfor a given circle, and its length is twice the length of a radius.

• Sector: a region bounded by two radii of equal length with a common center and eitherof the two possible arcs, determined by this center and the endpoints of the radii.

• Segment: a region bounded by a chord and one of the arcs connecting the chord’sendpoints. The length of the chord imposes a lower boundary on the diameter ofpossible arcs. Sometimes the term segment is used only for regions not containing thecenter of the circle to which their arc belongs to.

• Secant: an extended chord, a coplanar straight line, intersecting a circle in two points.

• A tangent: to a circle is a line in the plane of the circle which has exactly one commonpoint with the circle. The tangent to a circle is perpendicular to the radius passingthrough the point of contact .

Figure 2: Arc, sector, and segment

Figure 3: Chord, secant, tangent, radius, and diameter

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Definition 2. The line L is tangent to a circle if they have exactly one common point andall the other points on the line are external.

Example 1. Find the tangent lines to the parabola given by the graph of the function

f(x) = x2.

Solution :

Let x = 1 be fixed and consider the chord passing through the points

(1, 1) and (x, x2).

The slop

m(x) =x2 − 1

x− 1

is obviously depend on x. What happens if x tends to 1. Since the division by zero isimpossible we have to eliminate the term x− 1. Since

m(x) =x2 − 1

x− 1=

(x− 1)(x+ 1)

x− 1= x+ 1.

the slope at the limit position must be 2. The equation of the tangent line at x = 1 is

y − f(1) = 2(x− 1) ⇒ y = 2x− 1.

Tangents from an External Point

Theorem 3. Tangent segments from a common point external to a circle have the samelength.

Proof. Let O be the center of the circle, P the common endpoint of the tangent segments,and A and B their points of tangency. Then angles OAP and OBP are right angles (becausetangents are perpendicular to radii). Since segments OA and OB are radii, they are the samelength. OP is a common hypotenuse of right triangles OAP and OBP , so these trianglesare congruent by HL. Therefore AP = BP since they are corresponding parts.

Theorem 4. Let a circle with center O be given and suppose that P is an external point.The tangent lines from P to the circle can be constructed as follows:

• Draw a circle with radius R = OP/2 around the midpoint of OP .

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Figure 4: Tangent segments from an external point

• The circle constructed in the first step meets the given circle at two points A and B.

• AP and BP are tangent segments to the given circle.

To compute the common length of the tangent segments PA and PB we can use Pythagoreantheorem:

PA2 +R = OP 2.

Corollary 5. The tangent segments passing through a given external point are of the samelength.

Example 2. AP = x+6 and OP = 2x− 3 are tangent to circle O. Find the radius of circleO:

Solution : Since tangents from an external point are congruent, we can solve the equation

x+ 6 = 2x− 3

to find x = 9. Therefore PS = 9. Now draw the radius from O to A and let its length be R.Then OA = R and OS = R, and OA is perpendicular to AP .

Since triangle OAP is a right triangle, we can use the Pythagorean Theorem to find R:

R2 + 152 = (R + 9)2

R2 + 225 = R2 + 18R + 81

144 = 18R

R = 8

So the radius of circle O is 8 units.

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Figure 5: A tangential quadrilateral with its incircle

Tangential and cyclic quadrilaterals

Definition 6. In Euclidean geometry, a tangential quadrilateral (sometimes just tangentquadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all canbe tangent to a single circle within the quadrilateral. This circle is called the incircle ofthe quadrilateral or its inscribed circle, its center is the incenter and its radius is called theinradius. Since these quadrilaterals can be drawn surrounding or circumscribing their incir-cles, they have also been called circumscribable quadrilaterals, circumscribing quadrilaterals,and circumscriptible quadrilaterals. Tangential quadrilaterals are a special case of tangentialpolygons.

Theorem 7. The quadrilateral ABCD is a cyclic quadrilateral if and only if the sums ofthe opposite angles are equal.

Definition 8. A quadrilateral is called tangential if it has an inscribed circle which touchesall the sides of the quadrilateral.

Theorem 9. A convex quadrilateral is tangential if and only if the sum of the opposite sidesare equal.

Figure 6: Cyclic and tangential quadrilaterals

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The area of circles

The area of the unit circle

Everybody knows that the area of a circle with radius R is R2π. If we have a unit circlethen the area is just π.

The Circumference C = 2Rπ, and therefore, the radius R = C2π.

Also, the diameter D = 2R, and so, the radius R = D2.

4 Ways to Calculate the Area of a Circle :

• Using Radius to Find Area

For this example, assume that you are told that the radius of a given circle is 6 cm.

For the given example with a radius of 6 cm, the area is calculated as:

A = πR2

A = π62

A = 36π = 36(3.14) = 113.04 cm2

Figure 7: Case 1

• Calculating Area from the Diameter

Assume for this example that the diameter of your circle is 20 inches. The calculationsas follows:

D = 20

R =D

2=

20

2= 10

A = πR2

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A = π102

A = 100π = (100)(3.14) = 314. sq.in.

• Using Circumference to Calculate Area

For this example, assume that you have been told or have measured that the circum-ference of a circle (or circular object) is 42 cm.

A = πR2

A = π(C

2π)2

A = π(C2

4π2)

A =C2

4π

For C = 42 inches.

A =422

4π=

1764

4π=

441

π=

441

3.14= 140.4 cm2.

Case 2 Case 3

• Finding Area from a Sector of the Circle

Example 3. A sector of Circle O has an area of 15πcm2 and the central angle is 45◦

degrees . Find the area of Circle O.

When you know the area of a sector and its central angle measurement, you can usethe following modified formula to find the area of the circle:

Acir = Asec360

C

– Acir is the area of the full circle;

– Asec is the area of the sector;

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– C is the central angle measure.

Insert these into this formula and solve as follows:

Acir = 15π360

45

Acir = 15π(8)

Acir = 120π = 120(3.14) = 376.8 cm2.

Figure 8: Case 4

To compute the area of a circle we use an approximation based on inscribed regularn-gons. The area of a regular n-gon inscribed in the unti circle

n[sin 360◦

n

2].

Homework : A regular pentagon inscribed in a circle of radius 2 in. Find the area ofthe pentagon.

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