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    Circle Properties and Missing Values

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    Please help describe how to solve problems based on circle properties and circle rules.

    © BrainMass Inc. brainmass.com December 24, 2021, 11:42 pm ad1c9bdddf
    https://brainmass.com/math/circles/circle-properties-missing-values-595027

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    Solve problems and justify arguments about chords and lines tangent to circles

    Let's take a quick review on a few parts of a circle.
    Circle
    The locus of a point, which moves such that its distance from a fixed point always a constant.

    The fixed point is called its center and the constant distance is called its radius.
    The boundary of the circle is called its circumference.

    Chord
    A line segment whose end points lie on the circumference of the circle is called a chord.
    AB is the chord.

    Let's review the properties related to the chord.
    Property 1:
    In a circle, chords equidistant from the center are congruent.

    Property 2:
    In a circle, congruent chords are equidistant from the center.

    Property 3:
    A perpendicular line drawn from the center of a circle to a chord bisects the chord.

    Property 4:
    The line joining the center and the midpoint of a chord is perpendicular to the chord.

    Tangent
    A tangent to a circle is a line in the plane of the circle that intersects the circle in exactly one point.
    The line m is called the tangent line.

    Point of tangency
    The point where a circle and a tangent intersect is the point of tangency.
    The point A is called the point of tangency
    .
    Now let's review the properties related to the tangent lines.
    Property1:
    If a line is tangent to a circle, then the line is perpendicular to the radius to the point of tangency.

    Property 2:
    If a line in the plane of a circle is perpendicular to a radius at its endpoints on the circle, then the line is tangent to the circle.

    AB is tangent to the circle O.

    Property 3:
    The two segments tangent to a circle from a point outside the circle are congruent.

    That is, TP = TQ.

    Let's work out problems related to the chords and tangents to the circles.
    Example
    Find the length of the missing side OA when AB = 8 cm and OC = 3 cm.

    Given : AB = 8 cm and OC = 3 cm
    To find: Length of OA.
    Property: Perpendicular from the center of a circle to a chord bisects the chord.
    ==> AC = CB = 4 cm [ since AB = 8 cm]
    Let's find the length of OA using the Pythagorean theorem.
    In ∆OCA, OA2 = OC2 + CA2
    OA2 = 32 + 42
    = 9 + 16
    = 25
    OA = 5 cm
    The length of the missing side OA is 5 cm.

    Example
    Find the value of x.

    Given: OC = 18 and AC = 25
    To find: Length of DE.

    1)The perpendicular drawn from O to DE is F which equals 18.
    ==> OC = OF = 18 [Given:Perpendicular drawn from the center to the chords are equal]
    2) AC = CB [Property:Perpendicular drawn from the center bisects the chord]
    ==> AB = AC + CB
    AB = 25 + 25 = 50
    3) AB = DE [Property: Chords equidistant from the center are congruent]
    50 = DE
    The length of the chord DE is 50.

    Example
    Find the length of OC.

    Given: AC = CB = 18 and DE = 36
    To find : Length of OC

    The chords AB and DE have same length.
    i.e AB = DE = 36
    ==> OF = OC [Property:Equal chords of a circle are equidistant from the center]
    12 = OC
    The length of OC is 12.

    Example
    Find the measure of the angle P.

    Given: PQ and PR are the tangents.
    To find: m<P

    OQ and OR are the radius of the circle perpendicular to the tangents PQ and PR.
    ==> <Q and <R are the right angles.

    PQOR is a quadrilateral whose angle measures have a sum of 360 degrees.
    m<P + m<Q + m<R + m<O = 360°
    m<P + 90° + 90° + 122° = 360°
    m<P + 180° + 122° = 360° [90° + 90° = 180°]
    m<P + 302° = 360° [180° + 122° = 302°]
    m<P = 360° - 302°
    m<P = 58°
    The measure of the angle P is 58°.

    Example
    Determine the perimeter of the triangle ABC.

    Given: AF = 12 cm , DB = 14 cm, and CE = 7 cm
    To find: Perimeter of the triangle ABC.

    The circle with center O is inscribed in the triangle ABC.
    AD = AF = 12 cm
    [Property: The two segments tangent to a circle from a point outside the circle are congruent]
    BD = BE = 14 cm
    CF = CE = 7 cm

    We know that perimeter is the sum of the side lengths of the figure.
    Perimeter of the triangle ABC = AD + DB + BE + EC + CF + FA
    = 12 + 14 + 14 + 7 + 7 + 12
    = 66
    The perimeter of the triangle ABC is 66 cm.

    AB2 = EB2 + AE2
    AB2 = 52 + 262
    AB2 = 25 + 676
    AB2 = 701
    AB = 26.47
    The distance between the gears is about 26.5 in.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 11:42 pm ad1c9bdddf>
    https://brainmass.com/math/circles/circle-properties-missing-values-595027

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