Circumference

The circumference of a circle is the distance around the circle.  For a circumference of C units and a diameter of d units or a radius of r units, C =  πd or C = 2πr.

Angles and Arcs

A central angle is an angle whose vertex is at the center of a circle and whose sides are radii.  A central angle separates a circle into two arcs -- a major arc and a minor arc.

Some properties of central angles and arcs:

• The sum of the measures of the central angles of a circle with no interior points in common is 360 degrees.
• The measure of a minor arc equals the measure of its central angle.
• The measure of a major arc is 360 minus the measure of the minor arc.
• Two arcs are congruent if and only if their corresponding central angles are congruent.
• The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

Arc Length

An arc is part of a circle and its length is a part of the circumference of the circle.

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tester » 12pm - Jul 22, 2011

Find the length of an arc of a circle.

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Find the length of an arc of a circle.

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Arc Length of Circle

Arcs and Chords

Points on a circle determine both chords and arcs.  Several properties are related to points on a circle:

In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

If all the vertices of a polygon lie on a circle and the circle is circumscribed about the polygon.

Congruent arcs are arcs that have the same measure.

Congruent central angles have congruent chords.

Congruent chords have congruent arcs.

Congruent arcs have congruent central angles.

Properties of Diameters and Chords:

In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc.

In a circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

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tester » 12pm - Jul 22, 2011

If a radius is perpendicular to a chord, then it bisects the chord. ...

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If a radius is perpendicular to a chord, then it bisects the chord. Use this relationship to solve for missing lengths of segments.

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Radius Perpendicular to Chord in Circle

Inscribed Angle Theorem

An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle.

The measure of an inscribed angle is half the measure of its intercepted arc.

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tester » 12pm - Jul 22, 2011

Find the measure of an inscribed angle whose vertex is on the circle.

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Find the measure of an inscribed angle whose vertex is on the circle.

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Inscribed Angle Whose Vertex is On the Circle

Inscribed Angles

If inscribed angles of a circle intercept the same arc, then the angles are congruent.

An inscribed angle intercepts a semicircle if and only if the angle is a right angle.

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tester » 12pm - Jul 22, 2011

Find the measure of an inscribed angle that intercepts a semicircle.

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Find the measure of an inscribed angle that intercepts a semicircle.

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Inscribed Angle that Intercepts a Semicircle

Inscribed Polygon

An inscribed polygon is one whose sides are chords of a circle and whose vertices are points on the circle.

If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

Sectors

The area of a circle is found by using the formula A =  πr² .  A sector is a pie-shaped portion of the circle enclosed by 2 radii and the edge of the circle.  A central angle of a sector is an angle whose vertex is at the center of the circle and crosses the circle.  The area of the sector is proportional to the part that the central angle is of 360 degrees.

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tester » 12pm - Jul 22, 2011

Find the area of a sector of a circle.

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Find the area of a sector of a circle.

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Area of Sector of a Circle

Visit www.msdgeometry.com -  your ultimate internet resource for comprehensive information on High School Geometry, Algebra, and Trigonometry.

Tangents

A tangent to a circle intersects the circle in exactly one point, called the point of tangency.  There are three important relationships involving tangents:

If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is a tangent to the circle.

If two segments from the same exterior point are tangent to a circle, then they are congruent.

Circumscribed Polygons

When a polygon is circumscribed about a circle, all of the sides of the polygon are tangent to the circle.

Angles Formed by Intersections On or Inside a Circle

A line that intersects a circle in exactly two points is called a secant.  The measures of angles formed by secants and tangents are related to intercepted arcs.

If two secants intersect in the interior of a circle, then the measure of the angle formed is one-half the sum of the measure of the arcs intercepted by the angle and its vertical angle.

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tester » 12pm - Jul 22, 2011

Find the measure of an angle that is formed by the intersection of...

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Find the measure of an angle that is formed by the intersection of two chords inside a circle.

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Angle Formed by Intersection Inside Circle

If a secant and a tangent intersect at the point of tangency, then the measure of each angle formed is one-half the measure of its intercepted arc.

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tester » 1pm - Jul 22, 2011

Find the measure of an angle formed by the intersection of a secant...

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Find the measure of an angle formed by the intersection of a secant and tangent of a circle. The vertex lies on the circle itself.

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Angle Formed by Secant and Tangent of Circle

Angles Formed by Intersections Outside a Circle

If secants and tangents intersect outside a circle, they form an angle whose measure is related to the intercepted arcs.

If two secants, a secant and a tangent, or two tangents intersect in the exterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs.

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tester » 12pm - Jul 22, 2011

Find the measure of an angle formed by two segments intersection...

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Find the measure of an angle formed by two segments intersection outside a circle.

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Angle Formed by Intersection Outside Circle

Segments Intersecting Inside a Circle

If two chords intersect in a circle, then the products of the measures of the chords are equal.

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tester » 12pm - Jul 22, 2011

Find the lengths of segments intersection inside a circle. The...

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Find the lengths of segments intersection inside a circle. The products of the measures of the chords are equal.

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Length of Segments Intersecting Inside Circle

Segments Intersecting Outside a Circle

If secants and tangents intersect outside a circle, then two products are equal.

If two secant segments are drawn to a circle from an exterior point, then the products of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant segment and its external secant segment.

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tester » 12pm - Jul 22, 2011

Find the lengths of segments that intersect outside a circle.

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Find the lengths of segments that intersect outside a circle.

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Secant - Secant Product Theorem

If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.

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tester » 12pm - Jul 22, 2011

Find the lengths of segments formed by the intersection of a secant...

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Find the lengths of segments formed by the intersection of a secant and tangent of a circle.

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Secant - Tangent Product Theorem

Visit www.msdgeometry.com -  your ultimate internet resource for comprehensive information on High School Geometry, Algebra, and Trigonometry.

A circle consists of all points in a plane that are a given distance, called the radius, from a given point called the center.

A segment or line can intersect a circle in several ways:

• A radius is a segment with endpoints that are the center of the circle and a point of the circle.
• A chord is a segment whose endpoints lie on a circle.
• A secant is a line that intersects a circle at two points.
• A tangent is a line in the same plane as a circle that intersects the circle at exactly one point, called the point of tangency.

A radius and diameter also intersect circles.

Two coplanar circles that intersect at exactly one point are called tangent circles.

If two segments are tangent to a circle from the same external point, then the segments are congruent.

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tester » 12pm - Jul 22, 2011

Two segments that are tangent to the same circle are congruent to...

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Two segments that are tangent to the same circle are congruent to each other.

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Segments Tangent to Circle Are Congruent

Visit www.msdgeometry.com - your ultimate internet resource for comprehensive information on High School Geometry, Algebra, and Trigonometry.

Here are the notes from today's class:

Sector of a circle is the region of a circle bounded by a central angle and its intercepted arc

Segment of a circle is a region of a circle bounded by a chord and its arc.

The area of a segment can be found by subtracting the area of the triangle from the area of the sector (see example below).

Visit www.msdgeometry.com -  your ultimate internet resource for comprehensive information on High School Geometry, Algebra, and Trigonometry.

In this chapter, your child will learn about circles. In particular, your child will look at the lines, segments, and angles found inside and outside of circles.

Your child will first have to review the lines and segments that intersect circles.

A chord is a segment whose endpoints lie on a circle.

A secant is a line that intersects a circle at two points.

A tangent line is in the same plane as a circle and intersects that circle at only one point. This one point is called the point of tangency.

There are three ways your child can talk about pairs of circles. The circles can be congruent, concentric, or tangent.

Your child will learn to identify different parts of a circle.

Sector—region bounded by two radii of a circle and their intercepted arc

Arc length—distance along an arc measured in linear units

Inscribed angle—angle whose vertex is on a circle and whose sides contain chords of the circle

Please open the attached PDF to read the entire letter which includes the formula chart.

Attachments:

File	Description	File size
ch11 parent_letter.pdf		159 Kb

Click on the links to watch short tutorials and take interactive quizzes: