SPECIAL SEGMENTS IN TRIANGLES KEYSTONE GEOMETRY

2 SPECIAL SEGMENTS OF A TRIANGLE: MEDIAN 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.

WHERE THE MEDIANS MEET IN AN ACUTE TRIANGLE: THE CENTROID B A D E C F In the acute triangle ABD, figure C, E and F are the midpoints of the sides of the triangle. The point where all three medians meet is known as the “Centroid”. It is the center of gravity for the triangle.

FINDING THE MEDIANS: AN ACUTE TRIANGLE A B C A B C A B C

FINDING THE MEDIANS: A RIGHT TRIANGLE A B CA B CA B C

FINDING THE MEDIANS: AN OBTUSE TRIANGLE A B C

7 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.

ALTITUDES OF A RIGHT TRIANGLE B A D F In a right triangle, two of the altitudes of are the legs of the triangle.

legs inside In a right triangle, two of the altitudes are legs of the right triangle. The third altitude is inside of the triangle. ALTITUDES OF A RIGHT TRIANGLE A B CA B CA B C

ALTITUDES OF AN OBTUSE TRIANGLE In an obtuse triangle, two of the altitudes are outside of the triangle. B A D F I K

In an obtuse triangle, two of the altitudes are outside the triangle. For obtuse ABC: BD is the altitude from B CE is the altitude from C AF is the altitude from A ALTITUDE OF AN OBTUSE TRIANGLE A B C A B C A B CD E F

A B C A B C A B C DRAW THE THREE ALTITUDES ON THE FOLLOWING TRIANGLE:

A B C A B C A B C

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

SPECIAL SEGMENTS OF A TRIANGLE: PERPENDICULAR BISECTOR 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 J K

EXAMPLES: 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, is the perpendicular bisector. In the right ∆MLN, is the perpendicular bisector. In the isosceles ∆POQ, is the perpendicular bisector. E A B M L N AB R O Q P FINDING THE PERPENDICULAR BISECTORS