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Transcript
Geometry Chapter 5: Relationships in Triangles Objective: 5.1 Bisectors, -Identify and use perpendicular bisectors and angle bisectors in a triangle Medians and -Identify and use altitudes and medians in a triangle Altitudes Perpendicular bisector Angle bisector Median Altitude A perpendicular bisector of a side of a triangle is a line, segment, or ray that is perpendicular to the side and passes through its midpoint. (Draw a diagram) A segment, ray, or line which divides an angle into two congruent angles. (Draw a diagram) A median is a line segment that connects the vertex of a triangle to the midpoint of the opposite side. (Draw a diagram) An altitude of a triangle is a segment from a vertex to the line containing the opposite side and is perpendicular to that side. (Draw a diagram) 1 Examples Find the value of x and y. Find the value of x. Find x. Find x. Assignment 5.1 Part I page 242 #13-16 2 Construction 1: 1. On a sheet of Patty Paper, draw a triangle with three sides of different lengths. Label the triangle JKL. 2. Fold the paper to construct the angle bisectors of all three angles in triangle JKL 3. What do you notice? Concurrent lines Point of concurrency Incenter Two properties of angle bisectors are: 1) a point is on the angle bisector of an angle if and only if it is equidistant from the sides of the angle, and (2) the incenter of a triangle is equidistant from each side of the triangle. (Incenter Theorem) Example Find x and EF if BD is an angle bisector. 3 Construction 2: 1. On a sheet of Patty Paper, draw a triangle with three sides of different lengths. Label the triangle MNO. 2. Fold the paper to construct the medians of all three angles in triangle MNO. 3. What do you notice? (Copy your diagram to the right) Centroid Construction 3: 1. On a sheet of Patty Paper, draw a triangle with three sides of different lengths and no right angles. Label the triangle XYZ. 2. Fold the paper to construct the altitudes of all three angles in triangle XYZ. 3. What do you notice? (Copy your diagram to the right) Orthocenter 4 Construction 4: 1. On a sheet of Patty Paper, draw a triangle with three sides of different lengths and no right angles. Label the triangle LMN. 2. Fold the paper to construct the perpendicular bisector of all three angles in triangle XYZ. 3. What do you notice? Copy your diagram to the right) Circumcenter Two properties of perpendicular bisectors are: (1) a point is on the perpendicular bisector of a segment if and only if it is equidistant from the endpoints of the segment, and (2) the circumcenter of a triangle is equidistant from the three vertices of the triangle. (Circumcenter Theorem) Examples Lines a, b, and c are perpendicular bisectors of triangle PQR and meet at A. Find x, y, and z. 5 Centroid Theorem 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. Examples In triangleLMN, P, Q, and R are the midpoints of LM, MN, and LN, respectively. Find x, y, and z. Assignment 5.1 Part II page 242 #6, 17-26 6 Find x, y, and z. 5.2 Inequalities and Triangles Objective: -recognize and apply properties of inequalities to the relationships between angles and sides of a triangle Angle-Side If one side of a triangle is longer than another side, then the angle opposite the Relationships longer side has a greater measure than the angle opposite the shorter side. 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 lesser angle. In a triangle, if one side is longer than another side, then the angle opposite the Examples longer side is larger than the angle opposite the shorter side. a. For each triangle, list the angles in order from greatest to least. C 5 6 7 A b. B For each triangle, list the sides in order from longest to shortest. J 66° 50° Assignment 5.2 page 252 #29-34, 37-42,46-48 64° L 7 K 5.4 The Triangle Inequality Objective: -apply the triangle inequality theorem to determine whether a triangle can be drawn given specific side lengths Triangle Inequality The sum of the lengths of any two sides of a triangle is greater than the length of Theorem the third side. Examples Determine whether it is possible to draw a triangle with sides of the given measures. Write yes or no. a. 15, 12, 9 b. 23, 16, 7 The measures of two sides of a triangle are given. Between what two numbers must the measure of the third side fall? a. 9 and 15 b. 11 and 20 Assignment 5.4 page 264 #18-32 even Summary 8