Day 36 Triangle Segments and Centers

Today’s Agenda Triangle Segments Perpendicular Bisector Angle Bisector Median Altitude Triangle Centers Circumcenter Incenter Centroid Orthocenter

Perpendicular bisector A perpendicular bisector of a line segment is a) perpendicular to it, and b) bisects it. Theorem: If a point is on a perpendicular bisector of a line segment, then that point is equidistant from the endpoints of that segment. If CD is a  bisector of AB, then AC  BC. AB C D

Perpendicular bisector The converse is true: If a point is equidistant from the endpoints of a line segment, then that point is on the  bisector of the segment. If: AC  BC, AD  BD Then: C and D are on the perpendicular bisector of AB AB C D

Perpendicular bisector Each side of a triangle will have a perpendicular bisector. A perpendicular bisector will not necessarily connect to a vertex.

Angle Bisector Remember than an angle bisector is a line, ray, or segment that divides an angle into 2 congruent angles. Let’s construct an angle bisector.

Angle Bisector Theorem : If a point is on an angle’s bisector, then that point is equidistant from the two sides of the angle. Remember, the distance between a point and a line is perpendicular! A B C D

Angle Bisector Converse Theorem : If a point in the interior of an angle is equidistant from the two sides of the angle, then it is on the angle’s bisector. If: AD = BD; AD  AC; BD  BC Then: CD bisects  ACB A B C D

Angle Bisector An angle bisector of a triangle is a segment that divides one of its angles into two congruent pieces. The segment connects to the opposite side. Every point on the angle bisector of a triangle is equidistant from two of the triangle’s sides. Equidistant- the same distance from the sides. In the middle, creating 2 congruent segments.

Median A median of a triangle is a segment that connects a vertex to the midpoint of the opposite side. A median divides a triangle into two smaller triangles of equal area (although not necessarily congruent).

Median An angle bisector and a median usually are not the same line (except on an isosceles triangle). To construct a median, we need to be able to find the midpoint of a segment (the same as finding a perpendicular bisector).

Altitude The altitude of a triangle is the perpendicular distance from one of its bases to the opposite vertex. In other words, the altitude is a segment that is perpendicular to one side and reaches the point across from that side. The altitude doesn’t have to intersect the base itself, just the line containing the base. The length of the altitude is the height of the triangle.

Isosceles Triangles In an isosceles triangle, these different segments converge. The altitude (from the base to the vertex angle) is also an angle bisector. The altitude also bisects the base, which makes it a median. Since it is perpendicular to the base, as well as bisecting it, the altitude is also the perpendicular bisector. We can prove all of this with congruent triangles. In an equilateral triangle, these lines converge from each vertex.

Mid-segment A mid-segment of a triangle is a segment connecting the midpoints of two of its sides.

Mid-segment To construct a mid-segment, use the technique to find a midpoint on two of the triangle’s sides, then connect them.

Triangle Centers A triangle has many different centers (see list). We will study four: Circumcenter Incenter Centroid Orthocenter When three or more lines intersect, they are called concurrent lines. The point where concurrent lines intersect is known as a point of concurrency. The four listed points are all points of concurrency.

Circumcenter The circumcenter of a triangle is formed at the intersection of its three perpendicular bisectors. The circumcenter is the point that is equidistant from all three vertices of a triangle. Circumcenters don’t have to be inside of the triangle. In a right triangle, the circumcenter will be the midpoint of the hypotenuse. The circumcenter is the center of the triangle’s circumcircle, which is the circle that passes through all three of the triangle’s vertices.

Circumcenter

Incenter The incenter of a triangle is formed at the intersection of its three angle bisectors. The incenter is the point that is equidistant from the three sides of a triangle The incenter is always inside the triangle. The incenter is the center of the triangle’s incircle, which is the largest circle that you can draw inside of the triangle. It touches each of the three sides at one point.

Incenter

Centroid A centroid is the point formed when the three medians of a triangle intersect. The centroid is always inside of the triangle. The centroid is exactly two-thirds along the way of each median. In other words, the centroid divides each median into two parts, one of which is twice as long as the other. The centroid is the center of gravity of a triangle. This means that if you place a triangle on the tip of your pencil at the centroid, it should be perfectly balanced.

Orthocenter If you draw all three altitudes on a triangle, they intersect at the orthocenter. The orthocenter is not always inside the triangle. In obtuse triangles, the orthocenter is outside. In right triangles, the orthocenter will be the vertex of the right angle.

Other observations In an equilateral triangle, all four centers will be at the same point. Three of the centers – orthocenter, centroid, and circumcenter – will always be co-linear (they will form a straight line). The line they form is called the Euler line. Some pretty good applets that involve triangle centers can be found at