TOPIC 6: CIRCLES
Definition of Terms Circle, Chord, Radius, Diameter, Circumference, Arc, Sector, Centre and Segment of a CircleDefine circle, chord, radius, diameter, circumference, arc, sector, centre and segment of a circleA circle: is the locus or the set of all points equidistant from a fixed point called the center.Arc: a curved line that is part of the circumference of a circleChord: a line segment within a circle that touches 2 points on the circle.Circumference: The distance around the circle.Diameter: The longest distance from one end of a circle to the other.Origin: the center of the circlePi(π):A number, 3.141592…, equal to (the circumference) / (the diameter) of any circle.Radius: distance from center of circle to any point on it.Sector: is like a slice of pie (a circle wedge).Tangent of circle: a line perpendicular to the radius that touches ONLY one point on the circle.NB: Diameter = 2 x radius of circleCircumference of Circle = PI x diameter = 2 PI x radiusCentral AngleThe Formula for the Length of an ArcDerive the formula for the length of an arcCircumference of Circle = PI x diameter = 2 PI x radius where PI =?= 3.141592…The Central AngleCalculate the central angleA central angle is an angle formed by two intersecting radii such that its vertex is at thecenter of the circle.<AOB is a central angle. Its intercepted arc is the minor arc from A to B.
The Concept of Radian MeasureExplain the concept of radian measureRadians
are the standard mathematical way to measure angles. One radian is
equal to the angle created by taking the radius of a circle and
stretching it along the edge of the circle.The
radian is a pure mathematical measurement and therefore is preferred by
mathematicians over degree measures. For use in everyday work, the
degree is easier to work with, but for purely mathematical pursuits, the
radian gives better results. You probably will never see radian
measures used in construction or surveying, but it is a common unit in
mathematics and physics.
Radians to Degree and Vice VersaConvert radians to degree and vice versaThe
unit used to describe the measurement of an angle that is most familiar
is thedegree. To convert radians to degrees or degrees to radians, the
following relationship can be used.angle in degrees = angle in radians * (180/pi)So, 180 degrees = pi radiansExample 1Convert 45 degrees to radiansSolution45 = 57.32*radiansradians = 45/57.32radians = 0.785Most
often when writing degree measure in radians, pi is not calculated in,
so for this problem, the more accurate answer would beradians = 45
pi/180 = pi/4Example 2Convert pi/3 radians to degreedegrees = (pi/3) * (180/pi)degrees = 180/3 = 60°Angles PropertiesCircle Theorems of Inscribed AnglesProve circle theorems of inscribed anglesAninscribedangle
is formed when two secant lines intersect on a circle. It can also be
formed using a secant line and a tangent line intersecting on a circle. Acentral angle,
on the other hand, is an angle whose vertex is the center of the circle
and whose sides pass through a pair of points on the circle, therefore
subtending an arc.In this post, we explore the relationship between
inscribed angles and central angles having the same subtended arc. The
angle of the subtended arc is the same as the measure of the central
angle (by definition).
In
the first circle,is a central angle subtended by arc. Angleis an
inscribed angle subtended by arc. In the second circle,is an inscribed
angle andis a central angle. Both angles are subtending arc.What
can you say about the two angles subtending the same arc? Draw several
cases of central angles and inscribed angles subtending the same arc and
measure them. Use a dynamic geometry software if necessary. Are your
observations the same?In
the discussion below, we prove one of the three cases of the
relationship between a central angle and an inscribed angle subtending
the same arc.TheoremThe
measure of an angle inscribed in a circle is half the measure of the
arc it intercepts. Note that this is equivalent to the measure of the
inscribed angle is half the measure of the central angle if they
intercept the same arc.ProofLetbe
an inscribed angle andbe a central angle both subtending arcas shown in
the figure. Draw line. This forms two isosceles trianglesandsince two
of their sides are radii of the circle.
In
triangle, if we let the measure ofbe, then angleis also. By theexterior
angle theorem, the measure of angle. This is also similar to triangle.
If we let angle, it follows thatis equal to 2y. In effect, the measure of the inscribed angleand the measure of central anglewhich is what we want to prove.The Circle Theorems in Solving Related ProblemsApply the circle theorems in solving related problemsExample 3An arc subtends an angle of 200 at the center of the circle of radius25cm.Find the length of this arc.Solutionr =25cm, ?=20°
The length of the arc is 8.73cm.Example 4An arc of length 5cm subtends 50° at the center of the circle, what is theradius of the circle?l=5cm, ?=50°, r=?
Chord Properties of a CircleChord Properties of a CircleIdentify chord properties of a circleImagine
that you are on one side of a perfectly circular lake and looking
across to a fishing pier on the other side. The chord is the line going
across the circle from point A (you) to point B (the fishing pier). The
circle outlining the lake’s perimeter is called thecircumference. Achord
of a circleis a line that connects two points on a circle’s
circumference.To
illustrate further, let’s look at several points of reference on the
same circular lake from before. If each point of reference (i.e. duck
feeding area, picnic tables, you, water fountain, and fishing pier) were
directly on this lake’s circumference, then each line connecting a
point to another point on the circle would be chords.
- The line between the fishing pier and you is now chord AC
- The line between the water fountain and duck feeding area is now chord BE
- The line between you and the picnic tables is chord CD
If
we had a chord that went directly through the center of a circle, it
would be called adiameter. If we had a line that did not stop at the
circle’s circumference and instead extended into infinity, it would no
longer be a chord; it would be called asecant.The Theorem on the Perpendicular Bisector to a ChordProve the theorem on the perpendicular bisector to a chord.Proof of Theorem
The Theorem on Parallel ChordsProve the theorem on parallel chordsParallel chords in the same circle always cut congruent arcs. Parallel chords intercept congruent arcs.
- Construct a diameter perpendicular to the parallel chords.
- What does this diameter do to each chord? The diameter bisects each chord.
- Reflect across the diameter (or fold on the diameter). What happens to the endpoints?The
reflection takes the endpoints on one side to the endpoints on the
other side. It, therefore, takes arc to arc. Distances from the center
are preserved. - What have we proven? Arcs between parallel chords are congruent.
The Theorems on Chords in Solving Related ProblemsApply the theorems on chords in solving related problemsExample 5The figure is a circle with centreO. GivenPQ= 12 cm. Find the length ofPA.
Solution:The radiusOBis perpendicular toPQ. So,OBis a perpendicular bisector ofPQ.
Example 6The figure is a circle with centreOand diameter 10 cm.PQ= 1 cm. Find the length ofRS.
Solution:
Tangent PropertiesA Tangent to a CircleDescribe a tangent to a circleTangent
is a line which touches a circle. The point where the line touches the
circle is called the point of contact. A tangent is perpendicular to the
radius at the point of contact.Tangent Properties of a CircleIdentify tangent properties of a circleA
tangent to a circle is perpendicular to the radius at the point of
tangency. A common tangent is a line that is a tangent to each of two
circles. A common external tangent does not intersect the segment that
joins the centers of the circles. A common internal tangent intersects
the segment that joins the centers of the circles.Tangent TheoremsProve tangent theoremsTheorem 1If
two chords intersect in a circle, the product of the lengths of the
segments of one chord equal the product of the segments of the other.
Intersecting Chords Rule: (segment piece)×(segment piece) =(segment piece)×(segment piece)Theorem Proof:
Theorem 2:If
two secant segments are drawn to a circle from the same external point,
the product of the length of one secant segment and its external part
is equal to the product of the length of the other secant segment and
its external part.
Secant-Secant Rule: (whole secant)×(external part) =(whole secant)×(external part)Theorem 3:If
a secant segment and tangent segment are drawn to a circle from the
same external point, the product of the length of the secant segment and
its external part equals the square of the length of the tangent
segment.
Secant-Tangent Rule:(whole secant)×(external part) =(tangent)2Theorems Relating to Tangent to a Circle in Solving ProblemsApply theorems relating to tangent to a circle in solving problemsExample 7Two
common tangents to a circle form a minor arc with a central angle of
140 degrees. Find the angle formed between the tangents.SolutionTwo tangents and two radii form a figure with 360°. If y is the angle formed between the tangents then y + 2(90) + 140° = 360°y = 40°.The angle formed between tangents is 40 degrees.
