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geometry-lesson-5-circle-essentials
Circle Essentials
A circle is a set of all points in a plane that are equidistant to a given point called center. We have introduced the unit circle in trigonometry.
-The is a segment that joins the center of the circle to a given point on the circle. In the following diagram, we have segment OF representing a radius.
All of a circle are congruent.
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-Any line segment joining two points of a circle is called a of the circle.
is a chord.
-A is a line that contains a chord. Line
is a secant and contains
.
-A of the circle is a chord passing through the center of the circle.
si a chord passing through center
. It is a diameter.
Congruent circles have congruent radii.
-Concentric circles are coplanar circles having a common center.
-An angle having its vertex at the center of the circle is called a .
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A is a line in the plane of a circle that intersects the circle in exactly one point, called the
.
In the following figure, ,
and
are called tangent segment, tangent line and tangent ray.
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Inscribed and circumscribed circles:
When each side of a polygon is a tangent to a circle, the circle is said to be in the polygon.
By the same token, the polygon is said to .
Conversely, 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.
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Inscribed angle:
An inscribed angle is an angle with the vertex on the circle and the sides containing chords of that circle.
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Major and minor arcs
The major and minor arcs added complete the circumference of the circle. If they are equal, they are called semicircles.
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Tangents: Theorems and postulates
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Theorem:
If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency
Note:
We can prove that by remembering that the shortest distance between a point and a line is on the perpendicular to that line.
Corollary:
Tangents to a circle from a point are congruent.
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Theorem:
If a line in the plane of a circle is perpendicular to a radius at its outer endpoint, then the line is tangent to the circle
Arcs and angles: Theorems and postulates
Postulate: Arc addition
The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs.
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Theorem:
In the same circle or in congruent circles, two minor arcs are congruent if and only if their central angles are congruent.
Arcs and Chords: Theorems and postulates:
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Theorem:
In the same circle or in congruent circles, the congruent arcs have congruent chords and congruent chords have congruent arcs.
Theorem:
If a diameter or radius is perpendicular to a chord, then it bisects that chord and its arc.
Theorem:
In the same circle or in congruent circles, if two chords are equally distant from the center, then they are congruent and vice versa.
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Inscribed angles: Theorems and postulates
Theorem:
The measure of an inscribed angle is equal to the half of the measure of its intercepted arc.
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Corollary:
1.If two inscribed angles intercept the same arc, then the angles are congruent
2.An angle inscribed in a semicircle is a right angle
3.If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary
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Theorem:
The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc.
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Theorem:
The measure of an angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs
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Theorem:
The measure of an angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside a circle is equal to half the difference of the measures of the intercepted arcs
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Circles and lengths of segments
Theorem:
When two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord.
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Theorem:
When two secant segments are drawn to a circle from an external point, the product of one secant segment and its external segment equals the product of the other secant segment and its external segment.
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Theorem:
When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment is equal to the square of the tangent segment.
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Construction works
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