Winpossible New York, New York, US,

November 9, 2008

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This is a part of the interactive online tutorial for Geometry learning with the help of audio visual lessons. It has been created to help you learn the concept, perception with explanations and includes solution to practice questions from the topic of ‘Circles’. The mini-lessons here include: Getting started, Measuring Circles, Equation of a circle in standard form, Circles: arcs, chords, tangents, sector, Inscribed and Circumscribed Polygons.

- Mathematics > General
- Mathematics > Geometry

- Grade 9
- Grade 10
- Grade 11
- Grade 12

Identify and describe relationships among inscribed angles, radii, and chords.

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

(+) Construct a tangent line from a point outside a given circle to the circle.

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

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This is a part of the interactive online tutorial for Geometry learning with the help of audio visual lessons. It has been created to help you learn the concept, perception with explanations and includes solution to practice questions from the topic of ‘Circles’. The mini-lessons here include: Getting started, Measuring Circles, Equation of a circle in standard form, Circles: arcs, chords, tangents, sector, Inscribed and Circumscribed Polygons.

This resource has been contributed by Winpossible, and can also be accessed on their website by clicking here - Getting started.

In this mini-lesson you'll learn how to factor the binomials and polynomials. It is an important step in solving problems in a good number of algebraic applications. In factoring polynomials, we determine all the terms that were multiplied together to get the given polynomial. We then try to factor each of the terms we found in the first step. This continues until we can’t factor anymore. For Example, here is the complete factorization of the polynomial

In this mini-lesson you'll learn how to factor the binomials and polynomials. It is an important step in solving problems in a good number of algebraic applications. In factoring polynomials, we determine all the terms that were multiplied together to get the given polynomial. We then try to factor each of the terms we found in the first step. This continues until we can’t factor anymore. For Example, here is the complete factorization of the polynomial

x^{4} – 16 = (x^{2} + 4)(x + 2)(x – 2)

This FREE mini-lesson is a part of Winpossible's online course that covers all topics within Algebra I. Click on the video below to go through it. If you like it, you can buy our online course in Algebra I by clicking here.

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This resource has been contributed by Winpossible, and can also be accessed on their website by clicking here - Getting Started - Circles.

This mini-lesson introduces and walks you through the basic concepts of the circles. You'll learn it with the help of some examples, practice questions with solution,and using video explanation by the instructor that brings in an element of real-class room experience. You can see here the overview of the important basics of the circle, radius, chord etc.

A circle is the set of points that are equidistant from a special point in the plane. If the special point*O* is the center, it is called circle *O*. Some real-world examples of a circle are: a wheel, surface of a coin.

The radius is a line segment joining the center of the circle with a point on the circle. If any point*A* is on the circle with center *O*, then *OA* is the radius of circle. If you place two radii end-to-end in a circle, you would have the same length as one diameter. Thus the diameter of a circle is twice as long as the radius, E.g. a circle with 5 cms radius, will have 10 cms diameter.

A chord is a line segment joining two endpoints that lie on a circle. Further you may note that the diameter of a circle is the longest chord since it passes through the center and it can be stated every diameter is a chord, but not every chord is a diameter.

This mini-lesson introduces and walks you through the basic concepts of the circles. You'll learn it with the help of some examples, practice questions with solution,and using video explanation by the instructor that brings in an element of real-class room experience. You can see here the overview of the important basics of the circle, radius, chord etc.

A circle is the set of points that are equidistant from a special point in the plane. If the special point

The radius is a line segment joining the center of the circle with a point on the circle. If any point

A chord is a line segment joining two endpoints that lie on a circle. Further you may note that the diameter of a circle is the longest chord since it passes through the center and it can be stated every diameter is a chord, but not every chord is a diameter.

This FREE mini-lesson is a part of Winpossible's online course that covers all topics within Geometry. Click on the video below to go through it. If you like it, you can buy our online course in Geometry by clicking here.

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This resource has been contributed by Winpossible, and can also be accessed on their website by clicking here - Measuring Circles.

In this section you'll learn, basics and formulas for measuring circles, using explanation and examples. You know from earlier learnings that the circumference of a circle is given by:

*C* = 2?*r*

Now we walk through ways to measure angle in degrees and in radians for a circle. A degree is defined as 1/360 of a rotation of a radius about the center of a circle. Simply put, a circle is divided into 360 equal degrees, and a right angle (1/4th of the rotation) is 90°. It also gives the geometric definition of radians based on measuring distances, which states that the measure in radians is determined by the intersected arc length (*s*) divided by radius (*r*). It can be expressed as,

(radians) = arc length/radius =*s*/*r*

For example, 0.84 radians when converted to degrees, it comes to 48.13°.

To remember- formulas for working with angles in circles:

In this section you'll learn, basics and formulas for measuring circles, using explanation and examples. You know from earlier learnings that the circumference of a circle is given by:

Now we walk through ways to measure angle in degrees and in radians for a circle. A degree is defined as 1/360 of a rotation of a radius about the center of a circle. Simply put, a circle is divided into 360 equal degrees, and a right angle (1/4th of the rotation) is 90°. It also gives the geometric definition of radians based on measuring distances, which states that the measure in radians is determined by the intersected arc length (

(radians) = arc length/radius =

For example, 0.84 radians when converted to degrees, it comes to 48.13°.

To remember- formulas for working with angles in circles:

- Central Angle: A central angle is an angle formed by two intersecting radii such that its vertex is at the center of the circle.
- Inscribed Angle: An inscribed angle is an angle with its vertex "on" the circle, formed by two intersecting chords.
- 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 formed are equal.
- Angle Formed Outside of a Circle by the Intersection of, "Two Tangents" or "Two Secants" or "a Tangent and a Secant": Angle formed outside is equal to half the difference of intercepted arcs.

This FREE mini-lesson is a part of Winpossible's online course that covers all topics within Geometry. Click on the video below to go through it. If you like it, you can buy our online course in Geometry by clicking here.

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This resource has been contributed by Winpossible, and can also be accessed on their website by clicking here - Equation of a circle in standard form.

In this mini-lesson you'll learn, how to find the equation of a circle when it is placed in the coordinate plane. It will be done with the help of some examples, practice questions with solution, and watching video as well as explanation by the instructor in own handwriting.

The equation of a circle in standard form is (*x* - h)^{2} + (*y* - k)^{2} = r^{2}, where (h, k) is the center and r is the radius of a circle. Also, when the center of the circle is (0, 0), and the radius is r, then the equation of a circle is *x*^{2} + *y*^{2} = r^{2}. E.g. if the radius of a circle is 2 with the center at origin, then the equation of the circle is *x*^{2} + *y*^{2} = 4.

You will also find the explanation for finding the coordinates of the center and the radius of a circle from equation of a circle. E.g. to find the center and radius of the circle whose equation is (*x* - 5)^{2} + (*y* - 1)^{2} = 36, relate the equation to the standard form (*x* - h)^{2} + (*y* - k)^{2} = r^{2} and we get h = 5, k = 1 and r = ?36 = 6. Thus, center is (5, 1) and r = 6.

Given the center of a circle (3, -1) and radius 7; to find the equation of the circle we plug in these values in the equation in standard form (*x* - h)^{2} + (*y* - k)^{2} = r^{2}. It works out to (*x* - 3)^{2} + (*y* + 1)^{2} = 49.

In this mini-lesson you'll learn, how to find the equation of a circle when it is placed in the coordinate plane. It will be done with the help of some examples, practice questions with solution, and watching video as well as explanation by the instructor in own handwriting.

The equation of a circle in standard form is (

You will also find the explanation for finding the coordinates of the center and the radius of a circle from equation of a circle. E.g. to find the center and radius of the circle whose equation is (

Given the center of a circle (3, -1) and radius 7; to find the equation of the circle we plug in these values in the equation in standard form (

This FREE mini-lesson is a part of Winpossible's online course that covers all topics within Geometry. Click on the video below to go through it. If you like it, you can buy our online course in Geometry by clicking here.

Member Rating

Curriki Rating**'P'** - This is a trusted Partner resource P**'P'** - This is a trusted Partner resource

This resource has been contributed by Winpossible, and can also be accessed on their website by clicking here - Circles: arcs, chords, tangents, sector.

In this section, you’ll learn more parts to the circles; say an arc- major and minor, sector, segment, secant and tangent. It will be done with the help of some examples, practice questions with solution, using multimedia explanation by the instructor and own handwriting.

An arc is a part of the circumference of a circle. The longer arc is called the major arc while the shorter one is called the minor arc. Arc is measured in degrees and length. If the measure of minor arc is ?i.e. the measure of the central angle intercepted by the arc, then the measure of major arc is (360° - ). E.g. if measure of a minor arc is 100°, then major arc is (360° – 100°) = 260°. Now let us look at the relationship of radius and measure of central angle: if r is the radius of the circle and is the measure of central angle, then length of the arc intercepted by the angle ?is given by /360° × 2?r. E.g. = 120°, then length of the intercepted arc is 4?/3 units.

To remember-basic terms related to circle:

This FREE mini-lesson is a part of Winpossible's online course that covers all topics within Geometry. Click on the video below to go through it. If you like it, you can buy our online course in Geometry by clicking here.

In this section, you’ll learn more parts to the circles; say an arc- major and minor, sector, segment, secant and tangent. It will be done with the help of some examples, practice questions with solution, using multimedia explanation by the instructor and own handwriting.

An arc is a part of the circumference of a circle. The longer arc is called the major arc while the shorter one is called the minor arc. Arc is measured in degrees and length. If the measure of minor arc is ?i.e. the measure of the central angle intercepted by the arc, then the measure of major arc is (360° - ). E.g. if measure of a minor arc is 100°, then major arc is (360° – 100°) = 260°. Now let us look at the relationship of radius and measure of central angle: if r is the radius of the circle and is the measure of central angle, then length of the arc intercepted by the angle ?is given by /360° × 2?r. E.g. = 120°, then length of the intercepted arc is 4?/3 units.

To remember-basic terms related to circle:

- A chord is a straight line joining two points on the circumference. The longest chord is the diameter and diameter passes through the center. E.g.
*A*and*B*are the points on the circle. When we join these two points,*AB*is the chord. - A sector is a region enclosed by two radii and an arc. In a circle with center at
*O*and*A*,*B*two points on the circumference,*AOB*is the angle subtended by the arc*AB*at the centre*O*. The larger region is called the major sector and the smaller one, a minor sector. - A segment of a circle is the region enclosed by a chord and an arc of the circle. The larger segment is the major segment and the smaller one, the minor segment.
- A secant is a line intersecting the circle at two distinct points. If point
*A*and*B*are outside the circle and we join these, it intersects circle at two distinct points. - If a line and a circle have only one intersection point, this line is called a tangent. It is always perpendicular to the radius drawn to the point of intersection. This property is abbreviated as tan rad.
- The diameter of a circle divides the circle into equal halves and each part is called a semi circle.

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This resource has been contributed by Winpossible, and can also be accessed on their website by clicking here - Inscribed and Circumscribed Polygons.

In this mini-lesson, we'll discuss the concept of Inscribed and Circumscribed polygon and its properties. You will learn it with the help of some examples and practice questions with solution. Not that this might seem complicated here in text, but once you have instructor explain it to you in their voice and handwriting in the video, it would look much simpler.

An inscribed regular polygon is a polygon placed inside a circle such that each vertex of the polygon touches the circle. A circumscribed regular polygon is a polygon whose segments are tangent to a circle. Now take a look at and learn how to find out the area of inscribed and circumscribed polygons using the TA-CO method, in which the cut out(CO) is subtracted from the total area(TO) to get the required area. For example: in case of a circumscribed regular polygon, if a square floor of side 6m is covered by a circular shape rug, then the area of the floor uncovered by the rug is 6^{2} – ? x (3)^{2} = (36 – 9?) m^{2}.

This FREE mini-lesson is a part of Winpossible's online course that covers all topics within Geometry. Click on the video below to go through it. If you like it, you can buy our online course in Geometry by clicking here.

In this mini-lesson, we'll discuss the concept of Inscribed and Circumscribed polygon and its properties. You will learn it with the help of some examples and practice questions with solution. Not that this might seem complicated here in text, but once you have instructor explain it to you in their voice and handwriting in the video, it would look much simpler.

An inscribed regular polygon is a polygon placed inside a circle such that each vertex of the polygon touches the circle. A circumscribed regular polygon is a polygon whose segments are tangent to a circle. Now take a look at and learn how to find out the area of inscribed and circumscribed polygons using the TA-CO method, in which the cut out(CO) is subtracted from the total area(TO) to get the required area. For example: in case of a circumscribed regular polygon, if a square floor of side 6m is covered by a circular shape rug, then the area of the floor uncovered by the rug is 6

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