Download CIRCLES 10.1 Circles and Circumference CIRCLE

CIRCLES
10.1 Circles and Circumference
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CIRCLE - a set of points equidistant from one fixed point.
A circle is named by its center point.
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A chord is a segment with endpoints on the circle.
A diameter is a chord that passes through the center of the circle.
A radius is any segment with one endpoint at the center of the circle and the other
endpoint on the circle.
A radius is 1/2 the length of the diameter.
Circumference of a circle is the distance around the circle.
C = 2πr or C = πd
Perimeter - is the distance around a polygon
10.2 Angles and Arcs
A central angle of a circle has the center of the circle as its vertex, and its sides
are two radii of the circle.
The sum of the measures of the central angles of a circle with no interior points
in common is 360.
A central angle separates the circle into two parts – each of which is an arc.
The measure of each arc is related to the measure of its central angle.
A minor arc degree measure equals the central angle and is less than 180.
A major arc degree measure equals 360 minus the measure of the minor arc
and is greater than 180.
A semicircle is an arc and measure 180.
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In the same or congruent circles, two arcs are congruent if and only if their
corresponding central angles are congruent. Two angles are congruent if and only
if the corresponding central angles are congruent.
There are 2 ways to measure an arc:
1) Angles have degree measure, similarly, arc have degree measure. In a circle
graph, the central angles divide the circle into wedges often expressed by
percents.
The size of the angle is proportional to the percent. Multiply the
percent by 360 to determine the measure of the central angle.
2)
Arc Length is part of the length of the circumference of the circle. Use a
proportion.
Degree measure of arc
Degree measure of whole circle
=
arc length
circumference
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To find the length of the arc:
1. Find the circumference of the circle.
2. Divide the number of degrees in the arc by 360.
3. Then multiply the circumference of the circle by the ratio found in step 2.
10.3 Arcs and Chords
The endpoints of a chord are also endpoints of an arc.
Arcs and chords have a special relationship.
In a circle or congruent circles, two minor arcs are congruent if and only if their
corresponding chords are congruent.
In a circle or congruent circles, two chords are congruent if and only if they are
equidistant from the center of the circle.
The chords of adjacent arcs can form a polygon. This polygon is inscribed in the
circle because all of its vertices lie on the circle.
The circle circumscribes the
polygon.
Diameters that are perpendicular to chords create special segments and arc
relationships. In a circle, if a diameter or radius is perpendicular to a chord,
then it bisects the chord and its arc.
10.4 Inscribed Angles
An inscribed angle is an angle that has its vertex on the circle and its sides contained
in chords of the circle.
If an angle is inscribed in a circle, then the measure of the angle equals one-half of the
intercepted arc (or the measure of the angle).
Inscribed polygons also have special properties. An inscribed triangle with a side that
is a diameter is a special type of triangle.
If an inscribed angle intercepts a semicircle, the angle is a right angle.
If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
10.6 Tangents
A tangent intercepts a circle in exactly one point. This point is called
point of tangency.
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If a line is tangent to a circle, then it is
perpendicular o radius drawn to the point of tangency.
More than one line can be tangent to the same circle. If two segments from the same
exterior point are tangent to a circle, then they are congruent.
If a circle is inscribed in a polygon, then every side of the polygon is tangent to the
circle.
10.7 Secants, Tangents, and Angle Measures
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A line that intersects a circle in exactly two points is a secant.
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If two lines intersect a circle, there are three (3) places where the lines can intersect.
If a tangent and a chord intersect at a point ON a circle, then the
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measure of each angle formed is one half the measure of its
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intercepted arc.
If two chords intersect in the INTERIOR of a circle, then the measure of
the angle is 1/2 the sum of the measures of the arcs intercepted by the
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angle and its vertical angle.
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If a tangent and a secant, two tangents or two secants intercept in the EXTERIOR of
a circle, then the measure of the angle formed is one half the difference of the
measures of the intercepted arcs.
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There are several types of angles related to circles.
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The most popular ones are:
1. Central angles – vertex at center of circle - measure of angle = measure of arc
2. Inscribed angles – vertex on circle – measure of angle = ½ of intercepted arc
3. Two intersecting chords - vertex inside circle – measure of angle = ½ sum of 2 arcs.
4. Vertex outside of circle – measure of angle = ½ difference of 2 intercepted arcs.
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10.8 Special Segments in a Circle
When two chords intersect in the interior of a circle, each chord is divided into two
segments which are called segments of a chord. The following theorem gives a
relationship between the lengths of the four segments that are formed.
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If two chords intersect in the interior of a circle, then the product of
the lengths of the segments of one chord is equal to the product of the
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lengths of the segments of the other chord.
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If two secant segments share the same endpoint outside a circle, then the product of
the length of one secant segment and the length of its external segment
equals the product of the length of the other secant segment and the
length of its external segment.
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If a secant segment and a tangent segment share an endpoint outside a
circle, then the product of the length of the secant segment and the
length of its external segment equal the square of the length of the
tangent segment.
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10.9 Equations of Circles
You can write an equation of a circle in a coordinate plane if you know its radius and the
coordinates of its center.
Suppose the radius is r and the center is (h, k). Let (x, y) be any point on the circle.
The distance between (x, y) and (h, k) is r, so you can use the Distance Formula. (Told
you it wasn’t going away).
Square both sides to find the standard equation of a circle with radius r and center
(h, k).
( x  h) 2  ( y  k ) 2  r
(x – h)2 + (y – k)2 = r2
If the center is at the origin, then the standard equation is
(x – 0)2 + (y – 0)2 = r2
x2 + y2 = r2
Write the standard equation of the circle with a center at (- 4, 0) and radius 7.1
If you know the equation of a circle, you can graph the circle by identifying its center
and radius.
The equation of a circle is (x + 2)2 + (y - 3)2 = 9. Graph the circle.
First rewrite the equation to find the center and its radius.
To graph the circle, place the point of a compass at (- 2, 3), set the radius at 3 units,
and swing the compass to draw a full circle.