Chapter 10

10.1 The Circle

10.2 Congruent Chords

10.3 Arcs of a Circle

10.4 Secants and Tangents

Multiple Choice Review 10.1-4 2002

10.5 Angles Related to a Circle

10.6 More Angle - Arc Theorems

10.7 Inscribed and Circumscribed Polygons

10.8 Power Theorems

Chapter 10 Review 2002

Chapter 10 Review 2001

10.1 The Circle

Circle - definition – set of all points in a plane a given distance from a
given point.

Symbol - To distinguish between two circles we refer to them by their centers

Other definitions relating to circles

Center – the given point that the circle is equidistant from

Radius – the distance from the center to any point on the circle

Also refers to the segment that joins the center to any point on the circle.

Chord - a segment that joins any two points on a circle.

Diameter – any chord of a circle that contains the center.

Also refers the length of any chord that contains the center and is equal to two radii.

Tangent - A line that intersects a circle in exactly one point. The point

where the line intersects the circle is called the point of tangency.

Secant- A line that intersects a circle in two points.

Concentric Circles – circles with the same center but different radii

Interior of a circle – any point whose distance to the center of the circle is less than the circles radius is in the interior.

Exterior of a circle - A point is in the exterior of a circle if the distance to the center of the circle is greater than the radius of the circle.

F is in the interior of the circle. G is in the exterior of the circle.

Congruent Circles- circles whose radii are the same length.

Chord Theorems

If a radius is drawn perpendicular to a chord it bisects the chord (and its arc).

Introduce radii AB and AC, these are congruent because radii of a circle are congruent. If the angles formed at D are both right angles because of the definition of perpendicular, the triangles ADB and ADC are both right triangles. AD by reflexive is = to itself and the two triangles are congruent by HL. BD =DC because corresponding parts of congruent triangles are congruent.

Although we will learn this later on it is easy to prove additonally if the central angles in the circle are congruent, then their intercepted arcs are congruent.

Example 1

If the radius of circle J is 13 and chord MK is 5 cm from the center of the circle, find the length of the chord.

Solution: Draw in JM, use the pythagorean theorem to solve for MN (12 cm) then double your answer to find the length of MK = 24

Other chord theorems.

If a radius bisects a chord that is not a diameter, then it is perpendicular to the chord.

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

The last theorem is important if you are given a circle and asked to find its center.

Draw two non parallel chords of a circle. Construct the perpendicular bisectors of each of these. The intersection of the perpendicular bisectors is the center of the circle.

10.2 Other Chord Theorems

If two chords of a circle are congruent, then they are equidistant from the center of the circle.

The converse of this is also true.

If two chords of a circle are equidistant from the center of the circle, then the chords are congruent.

ADDITIONAL CHORD THEOREMS

Congruent arcs have congruent central angles

Congruent central angles have congruent arcs

Congruent arcs have congruent chords

Congruent chords have congruent arcs

10.3 Arcs of a circle (the central angle and arcs that were covered in 9.2 are repeated here)

Central Angle = an angle formed in a circle whose vertex is the center of the circle and whose sides are radii of the circle

An arc of a circle is the portion of a circle between two points, including these points.

A minor arc is an arc that is less than 1/2 of a circle. Its symbol is the two endpoints of the arc with a small eyebrow over the top as above.

A major arc is an arc that is greater than 1/2 of a circle. Its symbol is endpoint-point on arc- endpoint, with a larger eyebrow over the top. Below is an example of a major arc and its symbol.

The measure of an arc is defined to be = to the measure of the central angle that intercepts the arc.

Semi -circle- an arc that measures half of a circle

10.4 Secants and Tangents

Secant - a line that intersects a circle in two points

Tangent - a line that intersects a circle in exactly one point.

K is called the point of tangency

Tangent Rules

A tangent line is perpendicular to the radius drawn to its point of tangency.

Two Tangent Theorem (HAT Theorem) - If two tangent segments are drawn to a circle from an external point, then those segments are congruent.

Given: PA and PB are tangent segments from point P

Prove. PA = PB

Introduce OA and OB, OA=OB radii of a circle are congruent. OA and OB are perpendicular to the tangent segments (radii drawn to tangents at their point of tangency are perpendicular) Intro OP. The triangles are congruent POA=POB by HL so PA =PB by CPCTC

The reason it is referred to as the HAT theorem

Common Tangents to circles - lines that are tangent to more than one circle not necessarily at the same point

Common external tangents do not lie between the two circles ( do not intersect the segment joining the centers of the circles)

Common internal tangents lie between the circles (intersect the segment joining the centers of the circles)

Separate circles have 2 common external tangents and two common internal tangents Note that the two lines that intersect the dashed segment connecting the circles centers are the internal common tangents

10.5 Angles Related to a Circle

Central Angles are angles whose vertices lie at the center of a circle and whose sides are radii of the circle.

measure of a central angle = measure of its intercepted arc

Angles whose vertices are on the circle

Inscribed Angle - is an angle whose vertex is on the circle and whose sides are chords of the circle.

measure of an inscribed angle = measure of its intercepted arc.

Tangent Chord Angle - angle whose vertex is on the circle and whose sides are a chord and tangent to the circle.

measure of a tangent chord angle = 1/2 measure of its intercepted arc

Angles with vertices inside of a circle but not at the center.

Chord-chord angle - an angle formed by two chords intersecting inside a circle but not at the center

measure of a chord-chord angle = 1/2( sum of the intercepted arcs)

Angles with vertices outside a circle

Secant-secant angle - angle formed whose vertex is outside the circle and whose sides are secants to the circle

measure of a secant-secant angle = 1/2 ( subtraction of the intercepted arcs)

Secant-tangent angle- an angle formed by a secant and a tangent to a circle

measure of a secant-tangent angle =1/2 ( subtraction of the intercepted arcs)

Tangent-Tangent angle- an angle formed by two tangents drawn to a circle

measure of a tangent-tangent angle = 1/2(subtraction of the intercepted arcs)

**** Short cut -- the minor arc and the angle are always supplementary

10.6 More Angle-Arc Theorems

If two angles are inscribed on the same arc, then they are congruent.

This also applies to tangent chord angles that intercept the same arc.

Angles inscribed on semicircles are right angles

The only other rule in this section is the tangent-tangent angle shortcut which we already covered

10.7 Inscribed and circumscribed polygons

If a polygon is inscribed in a circle, all of its vertices lie on the circle

The circumcenter is the name for the center of the circle that is circumscribed about the polygon

If a polygon is circumscribed about a circle, all of its sides are tangent to the circle.

The incenter is the name for the center of the circle that is inscribed in the polygon.

Be able to construct each of the following

A circle circumscribed about a given triangle

To construct a circle about a triangle

Construct the perpendicular bisectors of two sides of the triangle

Use the intersection of these as the center and construct a circle whose radius = the distance to one of the three vertices.

A circle inscribed in a given triangle

Construct the bisectors of two of the angles

Drop a perpendicular from the intersection of the two angle bisectors

Use the perpendicular distance as the radius of the circle and the point of intersection as your center.

Tangents from an exterior point to a circle

Given Circle O with point P as an exterior point

Draw a segment OP

Construct the perpendicular bisector of OP

Using the midpoint of OP as the center and a radius of 1/2 OP, construct a circle. This should intersect both points P and O

Place points at the intersections of your two circles (A and B)

Draw in PA and PB

These are the tangents to the circle as they must be perpendicular to radii OA and OB (inscribed angles of the 2nd circle on semicircles)

*** Note - the only parallelogram that can be inscribed in a circle is a rectangle. ( remember squares are types of rectangles)

10. 8 Power Theorems

The power theorems deal with finding lengths of segments related to circles.

All of the power theorems can be proven by introducing lines and setting up proportions between similar triangles.

Chord-Chord Power Theorem

If two chords of a circle intersect, the product of the segments on one chord = the product of the segments on the second chord.

Notice that the two triangle formed by the introduction of the dashed lines, must be similar as they have two pair of angles inscribed on the same arc and also a pair of vertical angles. The proportion of sides would be a/c = d/b . A cross multiplication results in

(a) (b) = (c) (d)

Secant-Secant Power Theorem

If two secants are drawn from an exterior point to a circle, the product of the first secant segment and its external secant segment = the product of the second secant segment and its external secant segment

Again by introducing the dashed segments we end up with similar triangles where the proportiona sides are as follows

sec 1/sec 2 = ext 2/ ext 1

This results in the following formula

(SEC₁ ) (Ext₁) = (SEC₂ ) (Ext₂)

Tangent-Secant Power Theorem

If a tangent and secant are drawn to a circle from an exterior point, the tangent segment squared = secant segment times its external segment.

Notice that introduction of the dashed segments again forms similar triangles. Since the tangent is a side of both triangles, it becomes a geometric mean between the secant and its external segment.

sec/tan = tan/ ext

This results in the following formula

tan²= (sec) (ext)

Review

1. If chord AB measures 48 cm and is 7 cm from the center of the circle, what is the diameter of the circle.

2. Two circles intersect and have a common chord. If the centers of the circles are 14 inches apart and one of the circles has a radius of 15, and the other a radius of 13, find the length of the common chord

3. An equilateral triangle is inscribed in a circle with a radius of 6,

find the perimeter of the triangle.

4. If the measure of arc ABC = 240 degrees, the length of AC = 20 cm, O is the center of the circle and OM is perpendicular to AC, find OA length.

5. If AB is a common tangent, AC = 12, BD = 4 and CD = 16, find the

length of AB.

6. If Circle O is inscribed in a triangle with side lengths of 14, 16 and 19, the

length of x.

7. Given the concentric circles shown, with chord XY tangent to the inner circle, find the radius of the inner circle if the larger circle’s radius = 15 cm and XY = 12 cm.