7.1
Introduction and
Basic Terms
We considered circles much earlier in this course, primarily as a way to have problems in which there were some easily found congruent segments. Now we will develop some of the special attributes that circles and various related lines, angles, and arcs have. We start with some fundamental definitions.
- Definitions
Given a point of the plane, a circle is the set of all points of the plane that are the same distance from the given point. The given point is the center of the circle and the common distance is the radius of the circle.
We us the word "radius" in a second sense. It refers to any one of the segments whose endpoints are the center of the circle and a point of the circle. There is unlikely to be any confusion, since it is usually clear whether the radius under consideration is a number (a distance) or a segment. Points of the plane that are closer to the center of the circle than the radius lie in the interior of the circle. Points that are farther from the center than the radius lie in the exterior.
There are several other special terms that are used in relation to a circle. We have the following definitions.
- Definitions:
A chord of a circle is a
segment whose endpoints are two points of the circle.
A diameter of a circle is a chord that contains the center of the circle.
A secant is a line that contains two points of a circle.
A tangent to a circle is a line lying in the plane of the circle that intersects the circle in exactly one point.
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We say that two circles are congruent when they have equal radii. Two circles are concentric if the have the same cener but unequal radii. These situations are pictured below.
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Built into the definition of a circle is the fact that all radii of a circle are congruent. Since a circle consists of the points that lie at a particular distance from the center, the segments joining the center to the points on the circle are all the same length.
Given any point P of a plane, there are many circles passing through that point.
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Given any two points A and B of a plane, there are also many circles passing through those two points.
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- Theorem 7.1: There is exactly one circle that passes through any three noncolinear points.
Restatement: Given noncollinear points A, B, and C, one and only one circle exists that passes through all three of the points.
Given: A, B, and C are noncolinnear.
Prove: a. A, B, and C lie on a cirlce.
b. No second circle contains A, B, and C.
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Proof: (Part a) Since $\overline{AB}$ and $\overline{BC}$ are neither collinear nor parallel, their perpendicular bisectors must intersect. Call that point of intersection P. Since P is on the the perpendicular bisector of $\overline{AB}$, we have $PA=PB$. Since P is on the perpendicular bisector of $\overline{BC}$, we have $PB=PC$. Then $PA=PB=PC$, so that the three points A, B, and C are all the same distance from point P. Thus there is a circle that passes through the three noncollinear points A, B, and C (the center is at P).
(Part b) Now assume that the second circle also passes through the points A, B, and C. Let the center of this second circle be the point Q. Then $QA=QB=QC$. Sinc $QA=QB$, the point Q lies on the perpendicular bisector of $\overline{AB}$. Since $QB=QC$, the point Q lies on the perpendicular bisector of $\overline{BC}$. This means that the perpendicular bisectors of $\overline{AB}$ and $\overline{BC}$ intersect at Q as well as P. This is impossible, because two different lines cannot intersect in more than one point. Hence a second circle passing through A, B, and C is impossible.
When a circle passes through the vertices of a triangle, we say that the circle is circumscribed about the triangle and that the triangle is inscribed in the circle.
There are three very important relationships between chords and particular radii that we shall use frequently. These are detailed in the following three theorems.
- Theorem 7.2: In a circle, a radius that bisect a chord that is not a diameter is perpendicular to that chord.
- Theorem 7.3: A radius that is perpendicular to a chord of a circle bisects that chord.
- Theorem 7.4: The perpendicular bisector of a chord of a circle passes through the center of the circle.
The proofs of these theorems are left for the exercises. They suggest that in a problem involving a chord of a circle, not only are the radii drawn to the endpoints of the chord useful (since they give an isosceles triangle), but the radius drawn perpendicular to the chord ight come in handy, since it bisects the chord.