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Definition Concurrent Lines Are Lines That Intersect in a Single Point. 1. Theorem 128: the Perpendicular Bisectors of the Sides

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Definition Concurrent Lines Are Lines That Intersect in a Single Point. 1. Theorem 128: the Perpendicular Bisectors of the Sides 14.3 Notes Thursday, April 23, 2009 12:49 PM Definition 1. Concurrent lines are lines that intersect in a single point. j k m Theorem 128: The perpendicular bisectors of the sides of a triangle are concurrent at a point that is equidistant from the vertices of the triangle. This point is called the circumcenter of the triangle. D E F Theorem 129: The bisectors of the angles of a triangle are concurrent at a point that is equidistant from the sides of the triangle. This point is called the incenter of the triangle. A B Notes Page 1 C A B C Theorem 130: The lines containing the altitudes of a triangle are concurrent. This point is called the orthocenter of the triangle. A B C Theorem 131: The medians of a triangle are concurrent at a point that is 2/3 of the way from any vertex of the triangle to the midpoint of the opposite side. This point is called the centroid of the of the triangle. Example 1: Construct the incenter of ABC A B C Notes Page 2 14.4 Notes Friday, April 24, 2009 1:10 PM Examples 1-3 on page 670 1. Construct an angle whose measure is equal to 2A - B. A B 2. Construct the tangent to circle P at point A. P A 3. Construct a tangent to circle O from point P. Notes Page 3 3. Construct a tangent to circle O from point P. O P Notes Page 4 14.5 notes Tuesday, April 28, 2009 8:26 AM Constructions 9, 10, 11 Geometric mean Notes Page 5 14.6 Notes Tuesday, April 28, 2009 9:54 AM Construct: ABC, given {a, ha, B} a Ha B A b c B C a Notes Page 6 14.1 Notes Tuesday, April 28, 2009 10:01 AM Definition: A locus is a set consisting of all points, and only the points, that satisfy specific conditions. Four step procedure for locus problems 1. Find a single point that satisfies the given condition(s) 2. Find a second such point, and a third, and so on, until you can identify a pattern. 3. Look outside the pattern for points that you may have overlooked. Look within the pattern to exclude points that do not meet the conditions. 4. Present the answer by drawing a diagram and writing a description of the locus. Examples 1. Find the locus of points that are 1 inch from a given point. 2. What is the locus of all points equidistant from the sides of an angle. Notes Page 7 3. What is the locus of points 3 cm from a line . 4. What is the locus of points 2 cm from a given circle whose radius is 5 cm. 5. What is the locus of points less than 3 cm from a given point A. Notes Page 8 14.2 notes Tuesday, April 28, 2009 10:14 AM Compound Locus Many locus problems involve combining two or more loci in one compound locus. Compound Locus Procedure 1. Solve each part of the compound locus problem separately. 2. Find all the possible intersections of the loci. Examples: 1. If points A and B are 5 units apart, what is the locus of points 3 units from A and 4 units from B. 2. Find the locus of points that are a fixed distance from a given line and lie on a given circle. Step 1: Find each locus individually A. Locus of points that are a fixed distance from a given line. Notes Page 9 B. Locus of points that lie on a given circle. Step 2: Find all possible intersections of the loci. given line given line given line 4 points 3points 2 points given line given line 1 point no points empty set Solution: Empty set, 1 point, 2 points, 3 points or 4 points. 1. Find the locus in space that are a fixed distance from a given plane and a given distance from a fixed point on the plane. Notes Page 10 0 point s 2 points 2 circles Notes Page 11 .
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