Circles

CirclesNYU STEP geometry

Circle Basics

Def: A circle is the set of all points in a plane that are at the same distance from a fixed point called the center.

Def: The circumference of a circle is the distance around the circle. It contains 360 degrees.

Def: A radius of a circle is a line segment joining the center to a point on the circle.

Def: A central angle is an angle formed by 2 radii.

Def: An arc is a continuous part of a circle. *A semi-circle is an arc measuring one-half the circumference of a circle.

Def: A minor arc is an are that is less than a semi-circle. A major arc is an arc that is greater than a semi-circle.

Def: A chord is a segment that joins two points of the circle.

Def: A diameter of a circle is a chord through the center.

More Circle Basics

Area of a circle:

Circumference:

Circles and Chords

Theorems:

In a circle, a radius perpendicular to a chord bisects the chord.

In a circle, a radius that bisects a chord is perpendicular to the chord.

In a circle, the perpendicular bisector of a chord passes through the center of the circle.

Circles and Chords con’t.

Theorem: In a circle, or congruent circles, congruent chords are equidistant from the center.

(converse) In a circle, or congruent circles, chords equidistant from the center are congruent.

Circles and Chords con’t.

Theorem: In a circle, or congruent circles, congruent chords have congruent arcs.

(converse) In a circle, or congruent circles, congruent arcs have congruent chords.

Circles and Chords con’t

Theorem: In a circle, parallel chords intercept congruent arcs.

Tangents and Circles

Def: A tangent to a circle is a line in the plane of the circle that intersects the circle in exactly one point.

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

Tangents and Circles con’t.

Theorem: Tangent segments to a circle from the same external point are congruent.(You may think of this as the "Hat" Theorem because the diagram looks like a circle wearing a pointed hat.)

TWO DIFFERENT CIRCLES AND THEIR COMMON TANGENTS

Def: Common Tangent: If a line is a tangent to both the circles, then it is known as a common tangent to both the circles.

Def: Direct common Tangent :If the centers of the two circles lie on the same side of a common tangent to the circles, then it is known as Direct common tangent.

Def: Transverse common tangent: If the centers of the two circles lie on opposite sides of a common tangent to the circles, then it is known as Transverse common tangent .

Case(i):-Distance between the centers is greater than the sum of their radii:

Two Direct common tangents and two Transverse common tangents can be drawn.The two circles are non-intersecting type.

Case(ii):-Distance between the centers is equal to the sum of their radii:

In this case two Direct common tangents and one Transverse common tangent can be drawn. The two circles touch each other externally

Case(iii):-The distance between the centers is less than the sum and greater than the difference of their radii:Here the two circles are of intersecting type. Only two Direct common tangents can be drawn.

Case(iv):The distance between the centers is equal to the difference of their radii:

Here the two circles touch each other internally.

Only one transverse common tangent can be drawn.

Case (v):-The distance between the

centers is equal to zero: In this case one circle lies completely in the other.These are known as

Concentric circles

No common tangent can be drawn.

Rules for Dealing with Chords, Secants, Tangents in Circles Theorem: If two chords intersect in a circle, the product of the lengths

of the segments of one chord equal the product of the segments of the other.

Rules for Dealing with Chords, Secants, Tangents in Circles con’t. Theorem: If two secant segments are drawn to a circle from the same

external point, the product of the length of one secant segment and its external part is equal to the product of the length of the other secant segment and its external part.

Rules for Dealing with Chords, Secants, Tangents in Circles con’t. Theorem: If a secant segment and tangent segment are drawn to a

circle from the same external point, the product of the length of the secant segment and its external part equals the square of the length of the tangent segment.

Formulas for Angles in Circles Formed by Radii, Chords, Tangents, Secants

1. Central Angle: A central angle is an angle formed by two intersecting radii such that its vertex is at the center of the circle.


2. Inscribed Angle:An inscribed angle is an angle with its vertex "on" the circle, formed by two intersecting chords.


3. Tangent Chord Angle: An angle formed by an intersecting tangent and chord has its vertex "on" the circle.


Angle Formed Inside of a Circle by Two Intersecting Chords:When two chords intersect "inside" a circle, four angles are formed. At the point of intersection, two sets of vertical angles can be seen in the corners of the X that is formed on the picture. Remember: vertical angles are equal.


5. Angle Formed Outside of a Circle by the Intersection of:

"Two Tangents" or "Two Secants" or "a Tangent and a Secant".

The formulas for all THREE of these situations are the same:Angle Formed Outside = Difference of Intercepted Arcs

(When subtracting, start with the larger arc.)

Arcs in Circles

In a circle, the length of an arc is a portion of the circumference.

Arcs in Circles example

In circle O, the radius is 8, and the measure of minor arc AB is 110 degrees. Find the length of minor arc AB to the nearest integer.

Arcs and Circles

Theorem: In the same circle, or congruent circles, congruent central angles have congruent arcs.

Theorem: In the same circle, or congruent circles, congruent arcs have congruent central angles.

Arcs and Circles

Theorem: In the same circle, or congruent circles, congruent central angles have congruent chords.

Theorem: In the same circle, or congruent circles, congruent chords have congruent central angles.

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Circles