Keystone Geometry

» There are four types of segments in a triangle that create different relationships among the angles, segments, and vertices. ˃Medians ˃Altitudes ˃Angle Bisectors ˃Perpendicular Bisectors

3 Definition of a Median: A segment from the vertex of the triangle to the midpoint of the opposite side. Since there are three vertices in every triangle, there are always three medians.

»I»In acute, right and obtuse triangles the three medians are drawn inside the triangle. »T»To find the median, draw a line from the vertex to the midpoint of the opposite side. D D D

5 Special Segments of a triangle: Altitude Definition of an Altitude: The perpendicular segment from a vertex of the triangle to the segment that contains the opposite side.

AC B D »T»To find the altitude, draw a line from the vertex perpendicular to the opposite side. »I»In an acute triangle, the three altitudes are inside the triangle.

» In a right triangle, two of the altitudes are legs of the triangle and the third altitude is inside the triangle. » In an obtuse triangle, two of the altitudes are outside the triangle and the third altitude is inside the triangle. AC B

A B C A B C A B C

A B C A B C A B C

» Draw the three altitudes on the following triangle: A BC A BC A BC

» We already did this one in Unit 1 Part 1. » An angle bisector is a line, ray, or segment that divides an angle into two congruent smaller angles. » What about in a triangle? same thing! ANGLE BISECTOR THEOREM If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. If AD bisects BAC and DB = AB and DC = AC, then DB = DC

Solve for x. Because angles are congruent and the segments are perpendicular, then the segments are congruent. 10 = x + 3 x = 7 Because segments are congruent and perpendicular, then the angle is bisected which means they are are congruent. 9x – 1 = 6x x = 15 x = 3

perpendicular bisector The perpendicular bisector of a segment is a line that is perpendicular to the segment at its midpoint. The perpendicular bisector does NOT have to start at a vertex. In the figure, line l is a perpendicular bisector of JK. For a perpendicular bisector you must have two things: Show perpendicularity (90 degree angle) Show congruence (two equal segments) J K

Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If CP is the perpendicular bisector of AB, then CA = CB Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of the segment. If DA = DB, then D lies on the perpendicular bisector of CP.

KG = KH, JG = JH, FG = FH KG = KH 2x = x + 1 -x x = 1 GH = KG + KH GH = 2x + (x+1) GH = 2(1) + (1+1) GH = GH = 4

Draw the perpendicular bisector of the following lines, make one a ray, one a line, and one a segment. J K A B X Y

Example: C D In the scalene ∆CDE, AB is the perpendicular bisector. In the right ∆MLN, AB is the perpendicular bisector. In the isosceles ∆POQ, PR is the perpendicular bisector. E A B M L N AB R O Q P Remember, you must show TWO things. Show perpendicularity and congruence!