{"id":145,"date":"2017-08-23T09:10:08","date_gmt":"2017-08-23T09:10:08","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=145"},"modified":"2017-09-18T18:50:47","modified_gmt":"2017-09-18T18:50:47","slug":"geometric-proofs-using-coordinate-systems","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/geometric-proofs-using-coordinate-systems\/","title":{"rendered":"Geometric Proofs using Coordinate Systems"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/the-equation-of-a-circle\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/geometry-spatial-reasoning\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/measurement-and-linear-algebra\">Next Workshop \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Geometric Proofs using Coordinate Systems<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will use coordinate systems to prove geometric theorems.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>The diagonals of a parallelogram bisect each other.<\/li>\n<li>The diagonals of a rhombus are perpendicular.<\/li>\n<\/ul>\n<section>\n<h3><strong>How do we use coordinate geometry to prove that the\u00a0diagonals of a parallelogram bisect each other?<\/strong><\/h3>\n<p>The basic tools of coordinate geometry are the distance and\u00a0midpoint formulas and equations of lines, all of which were\u00a0discussed in previous lessons.<\/p>\n<p><span style=\"text-decoration: none;\">We can put any\u00a0parallelogram <\/span><em><span style=\"text-decoration: none;\">ABCD\u00a0<\/span><\/em><span style=\"text-decoration: none;\">in a plane, such that <\/span><em><span style=\"text-decoration: none;\">A\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is the origin and <\/span><em><span style=\"text-decoration: none;\">B\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is a point on the positive <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-axis,\u00a0i.e. <\/span><em><span style=\"text-decoration: none;\">A=<\/span><\/em><span style=\"text-decoration: none;\">(0,0) and <\/span><em><span style=\"text-decoration: none;\">B=<\/span><\/em><span style=\"text-decoration: none;\">(<\/span><em><span style=\"text-decoration: none;\">r,<\/span><\/em><span style=\"text-decoration: none;\">0)<\/span><em><span style=\"text-decoration: none;\">.\u00a0<\/span><\/em><span style=\"text-decoration: none;\">Let <\/span><em><span style=\"text-decoration: none;\">C=<\/span><\/em><span style=\"text-decoration: none;\">(<\/span><em><span style=\"text-decoration: none;\">u,v<\/span><\/em><span style=\"text-decoration: none;\">) and <\/span><em><span style=\"text-decoration: none;\">D=<\/span><\/em><span style=\"text-decoration: none;\">(<\/span><em><span style=\"text-decoration: none;\">s,t<\/span><\/em><span style=\"text-decoration: none;\">).\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s15_p2_html_m70793e4d.gif\" width=\"29\" height=\"23\" name=\"graphics16\" align=\"absmiddle\" border=\"0\" \/> is\u00a0parallel to\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s15_p2_html_299c8a9a.gif\" width=\"27\" height=\"21\" name=\"graphics17\" align=\"absmiddle\" border=\"0\" \/>,\u00a0so the points <\/span><em><span style=\"text-decoration: none;\">C\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">D\u00a0<\/span><\/em><span style=\"text-decoration: none;\">will have the same <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">-coordinate,\u00a0i.e., <\/span><em><span style=\"text-decoration: none;\">v=t<\/span><\/em><span style=\"text-decoration: none;\">.<br \/>\nFurthermore, since\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s15_p2_html_m3f2b1aa7.gif\" width=\"67\" height=\"23\" name=\"graphics18\" align=\"absmiddle\" border=\"0\" \/>\u00a0, <\/span><em><span style=\"text-decoration: none;\">u=r + s\u00a0<\/span><\/em><span style=\"text-decoration: none;\">. So, in fact, <\/span><em><span style=\"text-decoration: none;\">C=<\/span><\/em><span style=\"text-decoration: none;\">(<\/span><em><span style=\"text-decoration: none;\">r\u00a0+ s, t<\/span><\/em><span style=\"text-decoration: none;\">)<\/span><em><span style=\"text-decoration: none;\">.<\/span><\/em><\/p>\n<p align=\"CENTER\"><em><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.4%20Art%20019.JPG\" alt=\" Parallelogram ABCD in a plane\" width=\"218\" height=\"140\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/em><\/p>\n<p><span style=\"text-decoration: none;\">The diagonals are the line\u00a0segments\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s15_p2_html_m76b936f1.gif\" width=\"28\" height=\"23\" name=\"graphics19\" align=\"absmiddle\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s15_p2_html_m2baab905.gif\" width=\"27\" height=\"21\" name=\"graphics23\" align=\"absmiddle\" border=\"0\" \/>.\u00a0If <\/span><em><span style=\"text-decoration: none;\">P\u00a0<\/span><\/em><span style=\"text-decoration: none;\">denotes the point of their intersection, we want to show that <\/span><em><span style=\"text-decoration: none;\">P\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is the midpoint of both the segments\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s15_p2_html_m76b936f1.gif\" width=\"28\" height=\"23\" name=\"graphics20\" align=\"absmiddle\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s15_p2_html_m2baab905.gif\" width=\"27\" height=\"21\" name=\"graphics24\" align=\"absmiddle\" border=\"0\" \/>.\u00a0We can find <\/span><em><span style=\"text-decoration: none;\">P\u00a0<\/span><\/em><span style=\"text-decoration: none;\">by finding the equations of the lines containing\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s15_p2_html_m76b936f1.gif\" width=\"28\" height=\"23\" name=\"graphics21\" align=\"absmiddle\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s15_p2_html_m2baab905.gif\" width=\"27\" height=\"21\" name=\"graphics25\" align=\"absmiddle\" border=\"0\" \/>. <\/span><\/p>\n<p><span style=\"text-decoration: none;\">The slope <\/span><em><span style=\"text-decoration: none;\">m<sub>1\u00a0<\/sub><\/span><\/em><span style=\"text-decoration: none;\">of the line <\/span><em><span style=\"text-decoration: none;\">L<sub>1\u00a0<\/sub><\/span><\/em><span style=\"text-decoration: none;\">containing\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s15_p2_html_m76b936f1.gif\" width=\"28\" height=\"23\" name=\"graphics22\" align=\"absmiddle\" border=\"0\" \/>\u00a0is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s15_p2_clip_image002.gif\" width=\"41\" height=\"35\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/><\/span><em><span style=\"text-decoration: none;\">.\u00a0<\/span><\/em><span style=\"text-decoration: none;\">Recall that (<\/span><em><span style=\"text-decoration: none;\">y\u00a0\u2013 y<sub>1<\/sub><\/span><\/em><span style=\"text-decoration: none;\">)\u00a0<\/span><em><span style=\"text-decoration: none;\">= m<\/span><\/em><span style=\"text-decoration: none;\">(<\/span><em><span style=\"text-decoration: none;\">x\u00a0\u2013 x<sub>1<\/sub><\/span><\/em><span style=\"text-decoration: none;\">)\u00a0is the equation of the line with slope <\/span><em><span style=\"text-decoration: none;\">m\u00a0<\/span><\/em><span style=\"text-decoration: none;\">containing the\u00a0point (<\/span><em><span style=\"text-decoration: none;\">x<sub>1<\/sub>,\u00a0y<sub>1<\/sub><\/span><\/em><span style=\"text-decoration: none;\">).\u00a0Because <\/span><em><span style=\"text-decoration: none;\">L<sub>1\u00a0<\/sub><\/span><\/em><span style=\"text-decoration: none;\">contains the origin, the equation of <\/span><em><span style=\"text-decoration: none;\">L<sub>2\u00a0<\/sub><\/span><\/em><span style=\"text-decoration: none;\">is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s15_p2_clip_image004.gif\" width=\"68\" height=\"35\" name=\"graphics5\" align=\"TEXTTOP\" border=\"0\" \/>.\u00a0<\/span><span style=\"text-decoration: none;\">Similarly, the slope <\/span><em><span style=\"text-decoration: none;\">m<sub>2\u00a0<\/sub><\/span><\/em><span style=\"text-decoration: none;\">of the line <\/span><em><span style=\"text-decoration: none;\">L<sub>2\u00a0<\/sub><\/span><\/em><span style=\"text-decoration: none;\">containing <\/span><em><span style=\"text-decoration: none;\">BD <\/span><\/em><span style=\"text-decoration: none;\">is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s15_p2_clip_image006.gif\" width=\"41\" height=\"35\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>. Because <\/span><em><span style=\"text-decoration: none;\">L<sub>2\u00a0<\/sub><\/span><\/em><span style=\"text-decoration: none;\">contains <\/span><em><span style=\"text-decoration: none;\">B = <\/span><\/em><span style=\"text-decoration: none;\">(<\/span><em><span style=\"text-decoration: none;\">r,\u00a0<\/span><\/em><span style=\"text-decoration: none;\">0), the equation\u00a0of <\/span><em><span style=\"text-decoration: none;\">L<sub>2\u00a0<\/sub><\/span><\/em><span style=\"text-decoration: none;\">is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s15_p2_clip_image008.gif\" width=\"79\" height=\"38\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0.<br \/>\n<\/span><\/p>\n<p><span style=\"text-decoration: none;\">To find <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0set\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s15_p2_clip_image010.gif\" width=\"115\" height=\"38\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and solve for <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">.<br \/>\n<\/span><\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s15_p2_clip_image012.gif\" width=\"177\" height=\"224\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><span style=\"text-decoration: none;\">We can substitute <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">in either of the equations in order to find <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">.<br \/>\n<\/span><\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s15_p2_clip_image014.gif\" width=\"104\" height=\"38\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><span style=\"text-decoration: none;\">So the coordinates of <\/span><em><span style=\"text-decoration: none;\">P\u00a0<\/span><\/em><span style=\"text-decoration: none;\">are\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s15_p2_clip_image016.gif\" width=\"74\" height=\"40\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0If we apply the midpoint formula to <\/span><em><span style=\"text-decoration: none;\">A=<\/span><\/em><span style=\"text-decoration: none;\">(0,0) and <\/span><em><span style=\"text-decoration: none;\">C=<\/span><\/em><span style=\"text-decoration: none;\">(<\/span><em><span style=\"text-decoration: none;\">r+s, t<\/span><\/em><span style=\"text-decoration: none;\">), we get\u00a0<\/span><em><span style=\"text-decoration: none;\">M<sub>AC<\/sub>= <\/span><\/em><span style=\"text-decoration: none;\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s15_p2_clip_image017.gif\" width=\"74\" height=\"40\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0=\u00a0<\/span><em><span style=\"text-decoration: none;\">P<\/span><\/em><span style=\"text-decoration: none;\">.\u00a0Similarly, if we apply it to <\/span><em><span style=\"text-decoration: none;\">B=(r, 0) <\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">D=(s, t)<\/span><\/em><span style=\"text-decoration: none;\">, we get\u00a0<\/span><em><span style=\"text-decoration: none;\">M<sub>BD<\/sub><\/span><\/em><span style=\"text-decoration: none;\">=\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s15_p2_clip_image018.gif\" width=\"74\" height=\"40\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0=<\/span><em><span style=\"text-decoration: none;\">P<\/span><\/em><span style=\"text-decoration: none;\">.<\/span><em><span style=\"text-decoration: none;\"><br \/>\n<\/span><\/em><\/p>\n<h3><strong>How do we prove that the diagonals of a rhombus are\u00a0perpendicular?<\/strong><\/h3>\n<p>Since a rhombus is a\u00a0parallelogram, we can use exactly the same set-up that we used in\u00a0the previous proof.<\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.4%20Art%20020.JPG\" alt=\" Rhombus\" width=\"218\" height=\"140\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><span style=\"text-decoration: none;\">We need to show that\u00a0diagonals\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s15_p3_html_m76b936f1.gif\" width=\"28\" height=\"23\" name=\"graphics9\" align=\"absmiddle\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s15_p3_html_m2baab905.gif\" width=\"27\" height=\"21\" name=\"graphics6\" align=\"absmiddle\" border=\"0\" \/>\u00a0are perpendicular using the equations of the lines <\/span><em><span style=\"text-decoration: none;\">L<sub>1\u00a0<\/sub><\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">L<\/span><\/em><span style=\"text-decoration: none;\"><sub>2\u00a0<\/sub>containing them. <\/span><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What step must we show to finish the proof?<\/p>\n<ol>\n<li>Show that the midpoint of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s15_p3_html_m76b936f1.gif\" width=\"28\" height=\"23\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/>\u00a0is the midpoint of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s15_p3_html_m2baab905.gif\" width=\"27\" height=\"21\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/>.<\/li>\n<li><span style=\"text-decoration: none;\">Show that the slopes of <\/span><em><span style=\"text-decoration: none;\">L<sub>1\u00a0<\/sub><\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">L<\/span><\/em><span style=\"text-decoration: none;\"><sub>2\u00a0<\/sub>are reciprocals.<\/span><\/li>\n<li><span style=\"text-decoration: none;\">Show that the slopes of <\/span><em><span style=\"text-decoration: none;\">L<sub>1\u00a0<\/sub><\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">L<\/span><\/em><span style=\"text-decoration: none;\"><sub>2\u00a0<\/sub>are negative reciprocals.<\/span><\/li>\n<li><span style=\"text-decoration: none;\">Show that \u00a0<\/span><em><span style=\"text-decoration: none;\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s15_p3_html_m76b936f1.gif\" width=\"28\" height=\"23\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/>~<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s15_p3_html_m2baab905.gif\" width=\"27\" height=\"21\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/>.<br \/>\n<\/span> <span style=\"text-decoration: none;\"><br \/>\n<\/span><\/em><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is C . Two perpendicular lines have slopes\u00a0that are negative recipro<span style=\"text-decoration: none;\">cals.<\/span><strong><span style=\"text-decoration: none;\"><br \/>\n<\/span><\/strong><\/p>\n<\/div>\n<\/section>\n<h3>Review of New Vocabulary and Concepts<\/h3>\n<ul>\n<li><strong>Pythagorean Theorem : <\/strong><span style=\"text-decoration: none;\">The\u00a0square of the hypotenuse is equal to the sum of the squares of\u00a0the legs in any right triangle. <\/span><\/li>\n<li><strong>Converse of the\u00a0Pythagorean Theorem : <\/strong>i<span style=\"text-decoration: none;\">f\u00a0the sum of the squares of the lengths of two sides of a triangle\u00a0is equal to the square of the length of the third side, then the\u00a0triangle is a right triangle. <\/span><\/li>\n<li><span style=\"text-decoration: none;\">Given\u00a0a point P in the plane and let <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">be a positive\u00a0real number . The <\/span><em><span style=\"text-decoration: none;\">circle\u00a0<\/span><\/em><span style=\"text-decoration: none;\">with <\/span><em><span style=\"text-decoration: none;\">center\u00a0<\/span><\/em><em><span style=\"text-decoration: none;\">P\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and radius <\/span><em><span style=\"text-decoration: none;\">r <\/span><\/em><span style=\"text-decoration: none;\">is\u00a0the set of all points in the plane with distance <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">from <\/span><em><span style=\"text-decoration: none;\">P<\/span><\/em><span style=\"text-decoration: none;\">.<br \/>\n<\/span><\/li>\n<li>A <strong><em>chord\u00a0<\/em><\/strong>of a circle is a line segment with endpoints on the circle .<\/li>\n<li>The <strong><em>diameter <\/em><\/strong>of\u00a0a circle is length of a chord containing the center of the\u00a0<span style=\"text-decoration: none;\">circle and is denoted by <\/span><em><span style=\"text-decoration: none;\">d<\/span><\/em><span style=\"text-decoration: none;\">. The<\/span> diameter of a circle is twice its radius.<\/li>\n<li>An <em><strong>arc\u00a0<\/strong><\/em>is any connected part of a circle.<\/li>\n<li><span style=\"text-decoration: none;\">Any\u00a0two distinct points <\/span><em><span style=\"text-decoration: none;\">A\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">B <\/span><\/em>of\u00a0the circle divide it into two arcs called the <strong><em>minor\u00a0arc <\/em><\/strong><em>AB <\/em>and the <strong><em>major arc <\/em><\/strong><em>AXB<\/em>.<\/li>\n<li><strong>Arc Addition Theorem :<\/strong><span style=\"text-decoration: none;\">If\u00a0B is a point of the arc AC, then m(<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s15_p4_html_m21c79a1d.gif\" width=\"28\" height=\"24\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/>)\u00a0= m(<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s15_p4_html_m38d62e77.gif\" width=\"27\" height=\"23\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/>)\u00a0+ m(<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s15_p4_html_18d1ecc1.gif\" width=\"27\" height=\"24\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/>).<\/span><span style=\"text-decoration: none;\"><br \/>\n<\/span><\/li>\n<li><span style=\"text-decoration: none;\">The\u00a0measure of an inscribed angle is equal to half the measure of its\u00a0intercepted arc. <\/span><\/li>\n<li><span style=\"text-decoration: none;\">If\u00a0two inscribed angles intercept the same arc or congruent arcs,\u00a0then the angles are congruent.<\/span><span style=\"text-decoration: none;\"><br \/>\n<\/span><\/li>\n<li><span style=\"text-decoration: none;\">Two\u00a0arcs in the same circle or congruent circles are congruent if and\u00a0only if their corresponding chords are congruent. <\/span><\/li>\n<li>A <em><strong>secant <\/strong><\/em>of\u00a0a circle is any line intersecting the circle at two points.<\/li>\n<li>A <em><strong>tangent <\/strong><\/em>of\u00a0a circle is any line that intersects the circle at exactly one\u00a0point called the <strong><em>point of tangency<\/em><\/strong>.<\/li>\n<li><span style=\"text-decoration: none;\">A\u00a0line is tangent to a circle if and only if the radius drawn to\u00a0the point of tangency is perpendicular to the line. <\/span><\/li>\n<li><span style=\"text-decoration: none;\">The\u00a0measure of an angle formed by two secants that intersect in the\u00a0interior of a circle is one half the sum of the arcs intercepted\u00a0by the angle and its vertical angle. <\/span><\/li>\n<li><strong><span style=\"text-decoration: none;\">Arc\u00a0length formula: <\/span><\/strong><span style=\"text-decoration: none;\">s\u00a0= ar, where a is the arc measure and r is the radius of the\u00a0circle. <\/span><\/li>\n<li><span style=\"text-decoration: none;\">The\u00a0equation of a circle in the plane with center (<\/span><em><span style=\"text-decoration: none;\">h<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0<\/span><em><span style=\"text-decoration: none;\">k<\/span><\/em><span style=\"text-decoration: none;\">)\u00a0and with radius <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s15_p4_clip_image002.gif\" width=\"142\" height=\"17\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/><\/span><em><span style=\"text-decoration: none;\">.<br \/>\n<\/span><\/em><\/li>\n<li><span style=\"text-decoration: none;\">The standard form of\u00a0the equation of a circle is\u00a0<\/span><em><span style=\"text-decoration: none;\">x<sup>2<\/sup> + y<sup>2<\/sup> + Ax + By + C = 0. <\/span><\/em><\/li>\n<\/ul>\n<p><em>Geometry<\/em> (Ray Jurgenson and Richard G. Brown): Houghton\u00a0Mifflin, 1999.<\/p>\n<p><em>Geometry for Dummies<\/em> (Wendy Arnone): Wiley Publishing,\u00a0Inc., 2001.<\/p>\n<p><em>Schaum&#8217;s Outline of Geometry<\/em> (Barnett Rich, et. al.):\u00a0McGraw Hill, 1999.<\/p>\n<p><em>Standard Deviants: Geometry DVD 2-Pack<\/em> (Flavia Colonna\u00a0and Rebecca Berg): Cerebellum Corporation, 2000.<\/p>\n<p align=\"CENTER\">Don&#8217;t forget to test your knowledge\u00a0with the <a href=\"http:\/\/www.abcte.org\/drupal\/courses\/mrc\/quizzes\/geosr\" target=\"popsome\"> Geometry and Spatial Reasoning Chapter Quiz; <\/a><\/p>\n<\/section>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/the-equation-of-a-circle\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/geometry-spatial-reasoning\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/measurement-and-linear-algebra\">Next Workshop \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Workshop \u27a1 Geometric Proofs using Coordinate Systems Objective In this lesson, you will use coordinate systems to prove geometric theorems. Previously Covered: The diagonals of a parallelogram bisect each other. The diagonals of a rhombus are perpendicular. How do we use coordinate geometry to prove that the\u00a0diagonals of a parallelogram bisect [&hellip;]<\/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-145","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/145","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=145"}],"version-history":[{"count":11,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/145\/revisions"}],"predecessor-version":[{"id":766,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/145\/revisions\/766"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=145"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}