{"id":143,"date":"2017-08-23T09:09:12","date_gmt":"2017-08-23T09:09:12","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=143"},"modified":"2017-09-18T18:45:25","modified_gmt":"2017-09-18T18:45:25","slug":"circles","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/circles\/","title":{"rendered":"Circles"},"content":{"rendered":"
\n
\u2b05 Previous Lesson<\/a>\u00a0Workshop Index<\/a>\u00a0Next Lesson \u27a1<\/a><\/div>\n

<\/p>\n

Circles<\/h1>\n

Objective<\/h4>\n

In this lesson, we will prove the Pythagorean Theorem and its converse, and we will prove the formulas for special\u00a0right triangles that were covered in a previous section.<\/p>\n

\n

What are some basic definitions of the terms relating
\nto circles?<\/strong><\/h3>\n

Let <\/span>P\u00a0<\/span><\/em>be a point in the plane and let <\/span>r\u00a0<\/span><\/em>be a positive real number. The <\/span>circle<\/span><\/abbr> with <\/span>center<\/span><\/abbr> P\u00a0<\/span><\/em>and radius <\/span>r\u00a0<\/span><\/em>is the set of all points in the plane with distance <\/span>r\u00a0<\/span><\/em>from <\/span>P<\/span><\/em>.
\n<\/span><\/p>\n

A <\/span>radius<\/span><\/abbr> is any line\u00a0segment from the center <\/span>P\u00a0<\/span><\/em>to a point of the circle. <\/span><\/p>\n

\"Circle<\/p>\n

A <\/span>chord<\/span><\/abbr> of a circle is a line segment with endpoints on the circle. <\/span><\/p>\n

The <\/span>diameter<\/span><\/abbr> of a circle is\u00a0length of a chord containing the center of the circle and is\u00a0denoted by <\/span>d<\/span><\/em>.\u00a0The diameter of a circle is twice its radius. <\/span><\/p>\n

In the figure below, <\/span>AB<\/span><\/em>,\u00a0<\/span>CD<\/span><\/em>,\u00a0and <\/span>EF\u00a0<\/span><\/em>are all chords, but <\/span>AB\u00a0<\/span><\/em>is the only diameter shown. <\/span><\/p>\n

\"Circle<\/p>\n

How do we find the angle measure of an arc?<\/strong><\/h3>\n

An <\/span>arc<\/span><\/abbr> is any\u00a0connected segment of a circle.\u00a0<\/span>Any two distinct points <\/span>A\u00a0<\/span><\/em>and <\/span>B\u00a0<\/span><\/em>of the circle divide it into two arcs called the <\/span>minor arc<\/span><\/abbr> \u00a0<\/span>and the <\/span>major\u00a0arc<\/span><\/abbr><\/span>.\u00a0In the figure below, the colored arc\u00a0\u00a0is the minor arc\u00a0\u00a0and the black arc is major arc\u00a0<\/span>.
\n<\/span><\/p>\n

\"Major<\/p>\n

Each minor arc\u00a0<\/span>of a circle has an\u00a0associated central angle \u03b8<\/em> whose vertex is the center\u00a0<\/span>P\u00a0<\/span><\/em>and whose measure is less than 180\u00b0. <\/span><\/p>\n

If the measure of <\/span>\u03b8\u00a0<\/span><\/em>is 180\u00b0, then <\/span>A\u00a0<\/span><\/em>and <\/span>B\u00a0<\/span><\/em>are actually endpoints of a diameter and the two arcs are <\/span>semicircles<\/abbr>.<\/p>\n

T<\/span>he measure<\/abbr> of minor arc is the measure of its associated central\u00a0angle, the measure of a semicircle is 180\u00b0, and the measure of\u00a0a major arc is 360\u00b0 minus the measure of the corresponding\u00a0minor arc.<\/p>\n

Arc Addition Theorem <\/strong><\/h3>\n

If B is a\u00a0point of the arc AC, then m()\u00a0= m()\u00a0+ m().
\n<\/span><\/p><\/blockquote>\n

\n

Question<\/h4>\n
\n

What is the measure of the minor arc\u00a0in the figure below if\u00a0\u00a0= 155\u00b0 and m\u00a0EPD<\/span>=\u00a015\u00b0 ?<\/p>\n

\"<\/p>\n

    \n
  1. 130\u00b0<\/li>\n
  2. 140\u00b0<\/li>\n
  3. 155\u00b0<\/li>\n
  4. 160\u00b0<\/li>\n<\/ol>\n<\/div>\n

    Reveal Answer <\/a><\/p>\n

    \n

    The correct answer is B .\u00a0First note that\u00a0\u00a0=m<\/span>EPD\u00a0<\/span><\/em>=15\u00b0 . By the Arc Addition Theorem,\u00a0\u00a0<\/span>\u00b0<\/p>\n<\/div>\n<\/section>\n

    Circles<\/h3>\n

    An angle is <\/span>inscribed<\/span><\/abbr> in an arc if\u00a0the sides of the angles contain the endpoints of the arc and if\u00a0the vertex is a point on the circle touching the arc. In the\u00a0figure below, <\/span>ABC\u00a0<\/span><\/em>is inscribed in the major arc\u00a0\u00a0shown in color. <\/span><\/p>\n

    \"<\/p>\n

    An angle intercepts\u00a0an arc if the endpoints of the arc lie on the angle and each side\u00a0of the angle contains an endpoint of the arc. In the figure\u00a0below, each of the angles shown intercepts the arc\u00a0.<\/p>\n

    \"<\/p>\n

    Given an arc of a circle, what is the relationship of the\u00a0corresponding central angle to a corresponding inscribed angle?<\/p>\n

    \"relationship<\/p>\n

    The measure of\u00a0an inscribed angle is equal to half the measure of its intercepted\u00a0arc. <\/span><\/em><\/p><\/blockquote>\n

    We will prove a special case of this theorem, leaving the other\u00a0cases as an exercise.<\/p>\n

    Let <\/span>x\u00a0<\/span><\/em>denote the measure of the inscribed angle <\/span>ABC<\/span><\/em>.
    \n<\/span><\/p>\n

    There are three cases to consider:<\/p>\n

    Case 1:<\/strong> The inscribed\u00a0<\/span>ABC\u00a0<\/span><\/em>contains a diameter of the circle called <\/span>AB<\/span><\/em>.<\/span><\/p>\n

    Case 2:<\/strong> The\u00a0points <\/span>A\u00a0<\/span><\/em>and <\/span>C\u00a0<\/span><\/em>are on the same side of a diameter containing <\/span>B<\/span><\/em>.
    \n<\/span><\/p>\n

    Case 3:<\/strong> The\u00a0points <\/span>A\u00a0<\/span><\/em>and <\/span>C\u00a0<\/span><\/em>are on opposite sides of a diameter containing <\/span>B<\/span><\/em>.
    \n<\/span><\/p>\n

    Suppose we are in case 1 . Then\u00a0the triangle <\/span>PCB\u00a0<\/span><\/em>is an isosceles triangle, so m <\/span>ABC\u00a0<\/span><\/em>= m<\/span>\"angle\"PCB\u00a0<\/span><\/em>= <\/span>x<\/span><\/em>,\u00a0as shown below. <\/span><\/p>\n

    \"<\/p>\n

    Let <\/span>y\u00a0<\/span><\/em>=\u00a0denote\u00a0the measure of the intercepted arc <\/span>AC<\/span><\/em>.\u00a0The angles <\/span>APC\u00a0<\/span><\/em>and <\/span>CPB\u00a0<\/span><\/em>are supplementary, thus: <\/span><\/p>\n

    y <\/span><\/em>+\u00a0<\/span>z <\/span><\/em>=\u00a0180\u00b0. But, <\/span>x\u00a0<\/span><\/em>+ <\/span>x\u00a0<\/span><\/em>+ <\/span>z\u00a0<\/span><\/em>= 180\u00b0, so <\/span>y\u00a0<\/span><\/em>+ <\/span>z\u00a0<\/span><\/em>= 2<\/span>x\u00a0<\/span><\/em>+ <\/span>z<\/span><\/em>,\u00a0or <\/span>y\u00a0<\/span><\/em>= 2<\/span>x<\/span><\/em>. <\/span><\/p>\n

    \n

    Question<\/h4>\n
    \n

    What is the measure of the major arc\u00a0?<\/p>\n

    \"circle<\/p>\n

      \n
    1. 80\u00b0<\/li>\n
    2. 160\u00b0<\/li>\n
    3. 180\u00b0<\/li>\n
    4. 200\u00b0<\/li>\n<\/ol>\n<\/div>\n

      Reveal Answer <\/a><\/p>\n

      \n

      The correct answer is D. The measure of the intercepted\u00a0(minor) arc is equal to twice the measure of the inscribed angle,\u00a02(80) = 160\u00b0 . This means the measure of the major arc is 360\u00a0\u2013 160 = 200\u00b0.<\/p>\n<\/div>\n<\/section>\n

      Two circles are <\/span>congruent<\/span><\/abbr> if they have\u00a0congruent radii.<\/span><\/p>\n

      Two arcs are <\/span>congruent<\/span><\/abbr> if they lie on\u00a0the same circle (or on congruent circles) and have the same\u00a0measure. <\/span><\/p>\n

      The following two corollaries\u00a0essentially follow from the definitions and the previous theorem;\u00a0we will leave their proofs as an exercise.<\/p>\n

      If two\u00a0inscribed angles intercept the same arc or congruent arcs, then\u00a0the angles are congruent. <\/span><\/em><\/p>\n

      Two arcs in\u00a0the same circle or congruent circles are congruent if and only if\u00a0their corresponding chords are congruent. <\/span><\/em><\/p>\n

      What are tangent lines and secant lines?<\/strong><\/p>\n

      A <\/span>secant<\/span><\/abbr> of a circle is\u00a0any line intersecting the circle at two points. <\/span><\/p>\n

      A <\/span>tangent<\/span><\/abbr> of a circle is\u00a0any line that intersects the circle at exactly one point called\u00a0the <\/span>point\u00a0of tangency<\/span><\/abbr>.
      \n<\/span><\/p>\n

      A line is\u00a0tangent to a circle if and only if the radius drawn to the point\u00a0of tangency is perpendicular to the line. <\/span><\/em><\/p>\n

      \"Circle<\/em><\/p>\n

      There are a number of theorems on the angles formed by secant\u00a0and tangent lines, but we will just cover just two of them here.<\/p>\n

      The measure of\u00a0an angle formed by two secants that intersect in the interior of a\u00a0circle is one half the sum of the arcs intercepted by the angle\u00a0and by its opposite angle. <\/span><\/em><\/p>\n

      \"Circle<\/em><\/p>\n

      Let\u00a0\u00a0and\u00a0\u00a0denote the secants and Q\u00a0<\/span><\/em>the point of their intersection . Let\u00a01\u00a0be the measure of the angle formed by the secants. We need to show\u00a0that\u00a0.<\/p>\n

      Draw the chord\u00a0\u00a0and let\u00a02\u00a0= \u00a0\"angle\"QCB\u00a0<\/em>and\u00a0\"angle\"3\u00a0= \u00a0\"angle\"QBC<\/em>.<\/p>\n

      \"Circle<\/p>\n

      Because\u00a02\u00a0intercepts the arc\u00a0and\u00a03\u00a0intercepts the arc\u00a0\u00a0we have\u00a0\u00a0and\u00a0.\u00a0Furthermore, m\"angle\"1 = m\"angle\"2\u00a0+ m\"angle\"3.\u00a0This gives the desired result\u00a0.<\/p>\n

      The measure of\u00a0an angle formed by an intersecting secant and tangent is one half\u00a0the difference of the intercepted arcs<\/span><\/em>.<\/span><\/p>\n

      <\/p>\n

      Based on this therom, if\u00a0\"angle\"P\u00a0<\/em>denotes the angle formed by the intersection of secant AP\u00a0<\/span><\/em>and the tangent CP<\/span><\/em>,\u00a0then\u00a0\"angle\"<\/span>P=<\/span>.<\/em><\/p>\n

      How do we find the length of an arc?<\/strong><\/h3>\n

      Up to this point, we have only discussed the measure of arcs.\u00a0But since arcs are, in fact, segments, they also have a length.\u00a0Because an arc is curved, its length may seem more difficult to\u00a0measure.<\/p>\n

      If we use the formula for the circumference of a circle and set\u00a0up ratios, however, we can derive a formula for arc length.<\/p>\n

      The circumference of a\u00a0circle is equal to 2<\/span>\"pi\"r<\/span><\/em>, where <\/span>r\u00a0<\/span><\/em>is the radius of the circle. <\/span><\/p>\n

      Because the circumference\u00a0corresponds to an arc measure of 360 or 2<\/span>\"pi\"\u00a0<\/span><\/em>radians, the ratio of length <\/span>s\u00a0<\/span><\/em>of an arc to the circumference <\/span>C\u00a0<\/span><\/em>is equal to the ratio of the arc measure <\/span>a\u00a0<\/span><\/em>to 2<\/span>\"pi\"<\/span><\/em>.
      \n<\/span><\/p>\n

      <\/p>\n

      Arc length formula<\/strong><\/h3>\n

      s = ar, where\u00a0a is the arc measure and r is the radius of the circle.<\/p>\n

      \n

      Question<\/h4>\n
      \n

      What is the arc length in a circle of radius 3 corresponding to\u00a0a central angle of 60?<\/p>\n

        \n
      1. <\/li>\n
      2. <\/li>\n
      3. <\/li>\n
      4. 180\u00b0<\/li>\n<\/ol>\n<\/div>\n

        Reveal Answer <\/a><\/p>\n

        \n

        The correct answer is C. In order to use the formula, we need\u00a0to convert from degrees to radians.<\/p>\n

        radians
        \n=
        \n
        \nradians<\/p>\n

        By the formula for arc length,\u00a0.<\/p>\n<\/div>\n<\/section>\n<\/section>\n

        <\/p>\n

        \u2b05 Previous Lesson<\/a>\u00a0Workshop Index<\/a>\u00a0Next Lesson \u27a1<\/a><\/div>\n

        Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

        \u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Circles Objective In this lesson, we will prove the Pythagorean Theorem and its converse, and we will prove the formulas for special\u00a0right triangles that were covered in a previous section. What are some basic definitions of the terms relating to circles? Let P\u00a0be a point in the plane and […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-143","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/143","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=143"}],"version-history":[{"count":12,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/143\/revisions"}],"predecessor-version":[{"id":762,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/143\/revisions\/762"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=143"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}