Circles
-
Upload
orla-herring -
Category
Documents
-
view
10 -
download
0
description
Transcript of Circles
![Page 1: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/1.jpg)
Circles
![Page 2: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/2.jpg)
A circle is the set of all points in a plane that are equidistant from a given point called the center of the circle.
circle A, or A
A
![Page 3: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/3.jpg)
A
In a plane, the interior of a circle consists of the points that are inside the circle. The exterior of a circle consists of the points that are outside the circle.
interior
exterior
![Page 4: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/4.jpg)
Segments
and Lines
![Page 5: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/5.jpg)
The segment joining the center to a point on the circle is the radius of the circle.
Example:
Two circles are congruent if they have the same radius measures.
AG or AH
G
A
H
![Page 6: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/6.jpg)
A chord is a segment whose endpoints are points on the circle.
Ex: CD or EF
A
F
E
CD
![Page 7: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/7.jpg)
A diameter is a chord that passes through the center of the circle.
Ex: BC
A
C
B
![Page 8: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/8.jpg)
The distance across the circle, through its center, is the measure of the diameter of the circle. Ex: BC
A
C
B
![Page 9: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/9.jpg)
The length of a diameter is twice the length of a radius. The length of a radius is half the length of a diameter.
A
C
B
![Page 10: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/10.jpg)
A secant is a line that intersects a circle in two points
![Page 11: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/11.jpg)
A tangent is a line in the plane of a circle that intersects the circle in exactly one point. The point at which a tangent line intersects the circle to which it is tangent is the point of tangency.
![Page 12: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/12.jpg)
In a plane, two circles can intersect in two, one or no points. Coplanar circles that intersect in one point are called tangent circles.
![Page 13: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/13.jpg)
Coplanar circles that have a common center are called concentric.
A
![Page 14: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/14.jpg)
A line or segment that is tangent to two coplanar circles is called a common tangent.
![Page 15: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/15.jpg)
A common internal tangent intersects the segment that joins the centers of the two circles.
![Page 16: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/16.jpg)
A common external tangent does not intersect the segment that joins the centers of the two circles.
![Page 17: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/17.jpg)
TheoremsIf a line is tangent to a circle, then it is
perpendicular to the radius drawn to the point of tangency.
In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.
![Page 18: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/18.jpg)
Theorem
If two segments from the same exterior point are tangent to a circle, then they are congruent.
![Page 19: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/19.jpg)
Angles
and Arcs
![Page 20: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/20.jpg)
In a plane, an angle whose vertex is the center of a circle is a central angle of the circle.
A
C
Bcentral angle
![Page 21: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/21.jpg)
If the measure of a central angle is less than 180˚, then B and C and the points of A in the interior of BAC form a minor arc.
minor arc
A
C
B
![Page 22: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/22.jpg)
A
C
B
The measure of a minor arc is defined to be the measure of its central angle.
x x
![Page 23: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/23.jpg)
If the two endpoints of an arc are the endpoints of a diameter, then the arc is a semicircle.
A
C
BD semicircle
![Page 24: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/24.jpg)
The measure of a semicircle is 180˚.
A
C
BD semicircle
![Page 25: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/25.jpg)
A
C
B
If the measure of a central angle is greater than 180˚, then B and C and the points of A in the exterior of BAC form a major arc of the circle.
major arc
![Page 26: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/26.jpg)
The measure of a major arc is defined as the difference between 360˚ and the measure of its associated minor arc.
A
C
B
x(360 – x)
![Page 27: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/27.jpg)
Arc Addition Postulate
The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.
A
C
B
D
mBD mDC mBDC
![Page 28: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/28.jpg)
Inscribed
Angles
![Page 29: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/29.jpg)
An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc of the angle.
![Page 30: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/30.jpg)
Measure of an Inscribed Angle
If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.
![Page 31: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/31.jpg)
Theorem
If two inscribed angles of a circle intercept the same arc, then the angles are congruent.
![Page 32: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/32.jpg)
Properties of Inscribed Polygons
If all the vertices of a polygon lie on a circle, the polygon is inscribed in the circle and the circle is circumscribed about the polygon.
![Page 33: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/33.jpg)
Theorems About Inscribed Polygons
If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.
![Page 34: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/34.jpg)
Theorems About Inscribed Polygons
Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.
![Page 35: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/35.jpg)
A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.
![Page 36: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/36.jpg)
Other Angle Relationships in
Circles
![Page 37: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/37.jpg)
Theorem about Chords of Circles
In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.
![Page 38: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/38.jpg)
Theorem about Chords of Circles
If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
![Page 39: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/39.jpg)
Theorem about Chords of Circles
If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.
![Page 40: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/40.jpg)
Theorem about Chords of Circles
In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.
![Page 41: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/41.jpg)
Theorem
If a tangent and a secant intersect at a point on a circle, then the measure of each angle is one half the measure of its intercepted arc.
![Page 42: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/42.jpg)
If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
F
A
E
B
C
D
![Page 43: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/43.jpg)
12
m CFB mCB mDE
12
m DFE mCB mDE
F
A
E
B
C
D
![Page 44: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/44.jpg)
12
m CFD mCD mBE F
A
E
B
C
D
12
m BFE mCD mBE
![Page 45: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/45.jpg)
If a tangent and a secant intersect 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.
![Page 46: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/46.jpg)
12
m BVA mBA mCA
![Page 47: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/47.jpg)
If two tangents intersect 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.
![Page 48: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/48.jpg)
12
m BVA mBCA mBA
![Page 49: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/49.jpg)
If two secants intersect 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.
![Page 50: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/50.jpg)
12
m BVA mBA mDC
![Page 51: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/51.jpg)
Segment Lengths in Circles
![Page 52: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/52.jpg)
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 lengths of the segments of the other chord.
Ex: 8x = 24
x = 3
8 × 3 = 4 × 6
![Page 53: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/53.jpg)
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.
![Page 54: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/54.jpg)
Ex: 4(x + 4) = 3(3 + 5)
4x + 16 = 24
4x = 8
x = 2
4(2 + 4) = 3 (3 + 5)
outside segment × whole segment = outside segment × whole segment
![Page 55: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/55.jpg)
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 equals the square of the length of the tangent segment.
![Page 56: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/56.jpg)
Ex: 62 = 4(4 + x)
36 = 16 + 4x
20 = 4x
5 = x
62 = 4(4 + 5)
tangent squared = outside segment × whole segment
![Page 57: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/57.jpg)
Equations of Circles
![Page 58: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/58.jpg)
You can write an equation of a circle in a coordinate plane if you know its radius and the coordinate of its center.
![Page 59: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/59.jpg)
Suppose the radius of a circle is r and the center is (h, k). Let (x, y) be
any point on the circle.
![Page 60: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/60.jpg)
The distance between (x, y) and (h, k) is r, so you can use the
Distance Formula.
2 2( ) ( )r x h y k
![Page 61: Circles](https://reader036.fdocuments.net/reader036/viewer/2022062517/5681351a550346895d9c6fbe/html5/thumbnails/61.jpg)
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 2
If the center is the origin, then the standard equation is x 2 + y 2 = r 2