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Download Geometry 5-1 Bisectors, Medians, and Altitudes
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Transcript
Geometry 5-1 Bisectors, Medians, and Altitudes A. Special Segments 1. ________________________ __________________ -the perpendicular bisector does what it sounds like, it is perpendicular to a segment and it bisects the segment. -DF is a perpendicular bisector of AB in ABC C D A B F a.) Theorem 5-1 Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. b.) Theorem 5-2 Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment. c.) Since a triangle has three sides, there are three perpendicular bisectors in a triangle. The perpendicular bisectors of a triangle intersect at a common point. When three or more lines intersect at a common point, the lines are called ____________________ lines. d.) Their point of intersection is called the _______________ of __________________. e.) The point of concurrency of the perpendicular bisectors of a triangle is called the ___________. f.) Theorem 5-3 Circumcenter Theorem -The circumcenter of a triangle is equidistant from the vertices of the triangle. 2. ________________ _________________ -A segment that bisects an angle of the triangle and has one endpoint at a vertex of the triangle and the other endpoint at another point on the triangle. C A R B a.) Theorem 5-4 -Any point on the bisector of an angle is _____________ from the sides of the angle. b.) Theorem 5-5 -Any point on or in the interior of an angle and equidistant from the sides of an angle lies on the bisector of the angle. c.) The three angle bisectors in any triangle are concurrent and meet at the ___________________ of the triangle. d.) Theorem 5-6 Incenter Theorem -The incenter of a triangle is equidistant from each side of the triangle 3. Median -A segment that connects the vertex of a triangle to the midpoint of the side opposite that vertex. C A B F a.) The medians of a triangle also intersect at a common point, the point of _________________ for the medians of a triangle is called a centroid. b.) Theorem 5-7 Centroid Theorem -The centroid of a triangle is located twothirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median. 4. Altitude -It has one endpoint at a vertex of a triangle and the other on the line so that the segment is perpendicular to this line. C A D B a.) The intersection point of the altitudes of a triangle is called the orthocenter. Ex 1: ABC has vertices at A(-3, 10), B(9, 2), and C(9, 15). a.) Determine the coordinates of point P on AB so that CP is a median of b.) Determine if CP is an altitude of ABC . ABC. Chapter 5: Relationships in Triangles Section: 5.1 Bisectors, Medians, and Altitudes Vocabulary: _________________________________________: A line, segment, or ray that bases through the midpoint of a side of a triangle and is perpendicular to that side. _________________________________________: The intersection point of the three perpendicular bisectors of a triangle. _________________________________________: A segment whose endpoints are a vertex and the midpoint of the side opposite the vertex. _________________________________________: The intersection point of the three medians of a triangle. _________________________________________: A segment from a vertex to the opposite side and perpendicular to it. _________________________________________: The intersection point of the three altitudes of a triangle. Theorems: 5.1 Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. 5.2 Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment. 5.3 Circumcenter Theorem – The circumcenter of a triangle is equidistant from the vertices of the triangle. 5.4 Any point on the angle bisector is equidistant from the sides of the angle. 5.5 Any point equidistant from the sides of an angle lies on the angle bisector. 5.6 ________________________________________ – The incenter of a triangle is equidistant from each side of the triangle 5.7 ________________________________________ – The centroid of a triangle is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median. Section 5.2: Inequalities and Triangles Properties of inequalities for Real Numbers For all numbers a, b, and c a < b, a = b, or a > b ttttttttttttttttttttttttttttttttttttttt 1) If a < b and b < c, then a < c. 2) If a > b and b > c, then a > c. 1) If a > b and a + c > b + c, and a – c > b – c. 2) If a < b and a + c < b + c, and a – c < b – c. 1) If c > 0 and a < b, then ac < bc and a/c < b/c. 2) If c > 0 and a > b, then ac > bc and a/c >b/c. 3) If c < 0 and a < b, then ac > bc and a/c > b/c. 4) If c < 0 and a > b, then ac < bc and a/c < b/c. Theorems: 5.8 _______________________________________________ – If an angle is an exterior angle of a triangle , then its measure is greater than the measure of either of its corresponding remote interior angles. 5.9 If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. 5.10 If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the shorter angle. Section 5.3: Indirect Proof Steps for writing an Indirect Proof (Proof by Contradiction) 1) Assume that the conclusion is false. 2) Show that this assumption leads to a contradiction of the hypothesis, or some other fact, such as a definition, postulate, or corollary. 3) Point out that because the false conclusion leads to an incorrect statement, the original conclusion must be true. Section 5.4: The Triangle Inequality Theorems: 5.11 _______________________________________________ – The sum of the lengths of any two angle is greater than the length of the third side. 5.12 The perpendicular segment from a point to a line is the shortest segment form the point to the line. Corollaries: 5.1 The perpendicular segment from a point to a plane is the shortest segment from the point to the plane. Section 5.5: Inequalities Involving Two Triangles Theorems: 5.13 ______________________________________________– If two sides of a triangle are congruent to two sides of another triangle and the included angle in one triangle is has a greater measure than the included angle in the other triangle, then the third side of the first triangle is longer than the third side of the second triangle. 5.14 ______________________________________________– If two sides of a triangle are congruent to two sides of another triangle and the third side in one triangle is has a greater measure than the third side in the other triangle, then the angle between the pair of congruent sides in the first triangle is greater than the angle between the pair of congruent sides in the second triangle.