Geometry 5-1 Bisector of Triangles- Live lesson
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1 Geometry 5-1 Bisector of Triangles- Live lesson Draw a Line Segment Bisector: Draw an Angle Bisectors: Perpendicular Bisector A perpendicular bisector is a line, segment, or ray that is perpendicular to the given segment and passes through its midpoint. Concurrent Lines: When 3 or more lines intersect at a common point the lines Description Picture Perpendicular Bisector If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Converse of Perpendicular Bisector If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. Circumcenter Angle Bisector Converse of Angle Bisector Incenter The perpendicular bisectors of the sides of a triangle intersect at a point called the circumcenter that is equidistant from the vertices of the triangle. If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. If a point in the interior of an angle if equidistant from the sides of the angle, then it is on the bisector of the angle. The angle bisectors of a triangle intersect at a point called the incenter that is equidistant from the sides of the triangle. 1
2 Complete the problems and explain what theorem you are using. 1. BD is the perpendicular bisector of AC. Find x. Point P is the circumcenter of triangle ABC. List any segment(s) congruent to each segment below. 2. BR 3. CS Point A is the incenter of triangle PQR. Find each measure below. 4. <ARU 5. <QPK Find KF B C D 36 K 28 F 13 E 12 A 2
3 Geometry 5-2 Medians and Altitudes of Triangles- Live lesson Medians A median is a line segment that connects a vertex of a triangle to the midpoint of the opposite side. The three medians of a triangle intersect at the centroid of the triangle. The centroid is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median. Altitudes An altitude of a triangle is a segment from a vertex to the line containing the opposite side meeting at a right angle. Every triangle has three altitudes which meet at a point called the orthocenter. Example 1 The vertices of Triangle ABC are A(1, 3), B(7, 7) and C(9, 3). Find the coordinates of the orthocenter of Triangle ABC. Find coordinates of the orthocenter Step 1 Find the equation of one of the altitudes Step 2 Find the equation of another altitude Step 3 Solve the system of equations and find where the altitudes meet using substitution or elimination. 3
4 Example 2 Given triangle JHI with vertices J(1, 0), H(6, 0), I(3, 6) Find coordinates of the orthocenter Step 1 Find the equation of one of the altitudes Step 2 Find the equation of another altitude Step 3 Solve the system of equations and find where the altitudes meet using substitution or elimination. 4
5 Geometry 5-3 Inequalities in one triangle Exterior Angle : Exterior Inequality : Use the Exterior Angle Inequality to list all of the angles that satisfy the stated condition. 1. measures are less than m 1 2. measures are greater than m 5 3. measures are less than m 1 Angle Side Relationship 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. list the angle from greatest to least 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. list the sides from greatest to least 3. Find the shortest segment in the figure 4. List the side of Triangle MNO in order from longest to shortest if m<m=4x+20, m<n=2x+10 and m<o=3x
6 We are not doing 5-4 Geometry 5-5: The Triangle Inequality Triangle Inequality : Is it possible to form a triangle with given sides lengths? Explain why or why not 1. 3, 4, , 9, , 1, 5 Find the range for the measures of the third side of a triangle given the measures of two sides cm and 6 cm Find the range for the measure of the third side of a triangle given the measures of two sides cm and 6 cm yd and 18 yd Find the possible measures of x if each set of expression represents measure of the side of a triangle. x- 2, 10, 12 6
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