Intro. to Circles
Geometry
April 27, 2012
A circle is the set of points in a plane at a given distance from a given point in the
plane. The given point is the center of the circle and the given distance is the
radius (plural is radii). Any segment that joins the center to a point on the circle
is called a radius. Since the distance from the center to the points on a circle is
constant, all radii of a circle are congruent. A circle with its center at point 𝑃 is
denoted by ⨀𝑃.
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A chord is a segment whose endpoints lie on a circle. A secant is a line, ray or
segment that contains a chord. A diameter is a chord that contains the center of
a circle. Like the word radius, diameter can refer to the length of a segment, or
̅̅̅̅ is a chord, 𝐷𝐸
̅̅̅̅ is a diameter, and
to an actual segment. In the figure at right, 𝐵𝐶
⃡𝐹𝐺 is a secant.
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A tangent is a line, ray or segment in the plane of a circle that intersects the
circle in exactly one point, called the point of tangency. In the figure at right,
⃡ , ray 𝑃𝑇, and segment 𝑃𝑇
̅̅̅̅ may all be referred to as tangents to ⨀𝑂. 𝑇 is
line 𝑃𝑇
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the point of tangency.
A sphere is the set of points in space that are a given distance from a fixed
center point. All of the above terms for circles also apply to spheres.
Concentric circles.
Congruent circles (or spheres) are circles (or spheres) that have congruent
radii. Concentric circles are circles that lie in the same plane and have the
same center. The target at left illustrates concentric circles.
A polygon is inscribed in a circle and the circle is circumscribed about the
polygon when each vertex of the polygon lies on the circle, as illustrated in
the figures at left.
Triangle inscribed
in a circle.
Circle circumscribed
about a pentagon.
TANGENTS
Theorem: If a line is tangent to a circle (or sphere), then the line is
perpendicular to the radius drawn to the point of tangency. In the figure at
right, if line 𝑚 is tangent to ⨀𝑂 at 𝑇, then ̅̅̅̅
𝑂𝑇 ⊥ 𝑚.
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Proof: Assume temporarily that radius ̅̅̅̅
𝑂𝑇 is not perpendicular to tangent 𝑚.
Then the perpendicular segment from 𝑂 to 𝑚 intersects 𝑚 at some other point, say 𝑍. Since this perpendicular
segment from 𝑂 to 𝑚 is the shortest segment from 𝑂 to 𝑚, 𝑂𝑍 < 𝑂𝑇. By the definition of tangent, 𝑚 intersects ⨀𝑂
only at point 𝑇, so point 𝑍 must lie outside of ⨀𝑂, meaning that 𝑂𝑍 > 𝑂𝑇. But this is a contradiction, so our original
̅̅̅̅ ⊥ 𝑚.
assumption must be false, and it follows that 𝑂𝑇
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Corollary: Tangent segments to a circle from a common point are
congruent. In the figure at right, ̅̅̅̅
𝑃𝐴 and ̅̅̅̅
𝑃𝐵 are tangent to the circle at 𝐴
̅̅̅̅
̅̅̅̅
and 𝐵. By this corollary, 𝑃𝐴 ≅ 𝑃𝐵 . The proof of this is left as an exercise.
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Theorem: (Converse of the above theorem) If a line in the plane of a circle is perpendicular to
a radius at its outer endpoint, then the line is a tangent to the circle.
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̅̅̅̅ at 𝑅, then line 𝑘
In the figure at left, if line 𝑘, in the plane of ⨀𝑆, is perpendicular to radius 𝑅𝑆
is tangent to ⨀𝑆. The proof of this is also left as an exercise.
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When each side of a polygon is tangent to a circle, the polygon is said to be
circumscribed about the circle, and the circle is inscribed in the polygon.
A line that is tangent to each of two coplanar circles is called a common tangent.
Circumscribed polygons
and inscribed circles.
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A common internal tangent intersects the segment joining the centers; a common
external tangent does not intersect the segment joining the centers. In the figure
⃡ is a common internal tangent, and 𝐶𝐷
⃡ is a common external tangent.
at left, 𝐴𝐵
Tangent circles are coplanar circles that are tangent to the same line at the same
point. ⨀𝑋 and ⨀𝑌 are externally tangent, while ⨀𝑋 and ⨀𝑍 are internally
tangent.
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EXERCISES:
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1. In the figure at right, ̅̅̅
𝐽𝑇 is tangent to ⨀𝑂 at 𝑇.
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a. If 𝑂𝑇 = 6 and 𝐽𝑂 = 10, then 𝐽𝑇 = __________
b. If 𝑂𝑇 = 6 and 𝐽𝑇 = 10, then 𝐽𝑂 = __________
c. If 𝑚∠𝑇𝑂𝐽 = 60 and 𝐾𝑂 = 8, then 𝐽𝑂 = __________
d. If 𝐽𝐾 = 9 and 𝐾𝑂 = 8, then 𝐽𝑇 = __________
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2. The diagram at left shows tangent segments and circles. Find 𝑃𝐷.
PA = 8.2
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̅̅̅̅ and ̅̅̅̅
3. 𝑅𝑆
𝑇𝑈 are common internal tangents. If
𝑅𝑍 = 4.7 and 𝑍𝑈 = 7.3, find 𝑅𝑆.
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4. What do you think is true of common external
̅̅̅̅̅ and ̅̅̅̅
tangents 𝑊𝑋
𝑌𝑍? Will this be true if the
circles are congruent?
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̅̅̅̅ and ̅̅̅̅
5. Given: 𝑃𝐴
𝑃𝐵 are tangents to ⨀𝑂. Use the diagram at right to explain how the
corollary on the first page follows from the theorem that precedes it.
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6. In the same diagram, which pairs of angles are congruent? Which pairs of angles are complementary? Which
pairs of angles are supplementary?
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̅̅̅ are tangents to ⨀𝑂 from 𝑇; 𝑚∠𝑅𝑇𝑆 = 36.
̅̅̅̅ and ̅𝑇𝑆
7. Given: 𝑇𝑅
a. Find 𝑚∠𝑇𝑆𝑅 and 𝑚∠𝑇𝑅𝑆.
b. Find 𝑚∠𝑂𝑆𝑅 and 𝑚∠𝑂𝑅𝑆.
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c. Find 𝑚∠𝑅𝑂𝑆. Does your result agree with one of your conclusions from problem #6?
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̅̅̅̅ ⊥ 𝑃𝑂
̅̅̅̅.
̅̅̅̅ is tangent to ⨀𝑂 at 𝑇; 𝑇𝑆
8. Given: 𝑃𝑇
a. 𝑇𝑆 is the geometric mean between ________ and ________.
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b. 𝑇𝑂 is the geometric mean between ________ and ________.
c. If 𝑂𝑆 = 6 and 𝑆𝑃 = 24, then 𝑇𝑆 = ___________ and 𝑇𝑃 = ___________.
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̅̅̅̅ is a common internal tangent to ⨀𝐴 and ⨀𝐵.
9. Given: 𝑅𝑆
Explain why
𝐴𝐶
𝐵𝐶
=
𝑅𝐶
𝑆𝐶
.
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10 Quad. 𝐴𝐵𝐶𝐷 is circumscribed about a circle as shown at left. Discover and prove a
relationship between 𝐴𝐵 + 𝐷𝐶 and 𝐴𝐷 + 𝐵𝐶.
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̅̅̅̅ is tangent to ⨀𝑃 and ⨀𝑄; 𝑄𝑇 = 6; 𝑇𝑅 = 8; 𝑃𝑅 = 30.
11. 𝑆𝑅
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a. 𝑃𝑄 = ____________
b. 𝑃𝑆 = ____________
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c. 𝑆𝑇 = ____________
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12. ̅̅̅
𝐽𝐾 is tangent to ⨀𝑃 and ⨀𝑄. Find 𝐽𝐾.
(Hint: what kind of quad. is 𝐽𝑃𝑄𝐾?)
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13. Circles 𝑃 and 𝑄 are tangent to each other and have radii of 6 and 2.
Find the length of their common external tangent ̅̅̅̅
𝐴𝐵 .
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14. Given: Two tangent circles; ̅̅̅̅
𝐸𝐹 is a common external tangent; ̅̅̅̅
𝐺𝐻 is the
common internal tangent.
a. Discover and prove something interesting about point 𝐺.
b. Discover and prove something interesting about ∠𝐸𝐻𝐹.
15. The isosceles triangle at right has sides of length 12, 12, and 8.
Find the radius of the inscribed circle.