SEGMENTS IN TRIANGLES TRIANGLE GEOMETRY

2 SPECIAL SEGMENTS OF A TRIANGLE: MEDIAN Definition: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: 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

MIDSEGMENT THEOREM The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long.

THRM 5-8 PG 177, THRM 5-9 THRM 5-10 PG 178, THRM 5-11 (MIDSEGMENT THRM)

PROOFS ON PG 177, 178, Classroom exercises on pg Written exercises on page 180 Extra examples

PROOFS ON PG 177, 178, Classroom exercises on pg 179 2) 5 3) 14 4) 8 5).5k 6) A) 5 7 4, b c) 5 7 4, D )The segment joining the midpoints of the sides of a triangle divide the triangle into 4 congruent triangles 8)3 9)Thrm 5-8 (If two lines are parallel then all points on one line are equidistant from the other line. Written exercises on page 180 Extra examples