Day 36Triangle Segments and Centers

Today’s AgendaTriangle Segments

Perpendicular BisectorAngle BisectorMedianAltitude

Triangle CentersCircumcenterIncenterCentroidOrthocenter

Perpendicular bisectorA 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.Proof:

A B

C

D

Perpendicular bisectorThe converse is true: If a point is equidistant from

the endpoints of a line segment, then that point is onthe bisector of the segment.

Write this proof in groups.Given: AC BC, AD BD Prove: C and D are on the perpendicular

bisector of ABHint: Use a larger pair of congruent

triangles to prove that a smallerpair of triangles are congruent.A B

C

D

Perpendicular bisectorEach side of a triangle will have a

perpendicular bisector.A perpendicular bisector will not necessarily

connect to a vertex.

Perpendicular bisectorSee the following video for constructing a

perpendicular bisector.

Angle BisectorRemember than an angle bisector is a line,

ray, or segment that divides an angle into 2 congruent angles.

Let’s recall how to construct an angle bisector (see video).

Angle BisectorTheorem: 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!

Proof: A

B

C

D

Angle BisectorConverse 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.

Write the proof as a group.Given: AD = BD; AD AC; BD BCProve: CD bisects ACB A

B

C

D

Angle BisectorAn 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 triangleis equidistant from two of the triangle’s sides.

MedianA 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).Why is this?

MedianAn angle bisector and a median usually are

not thesame 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).

AltitudeThe 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.

AltitudeComplete the handout on altitudes.To construct an altitude, we need to know

how to create perpendicular lines. (See video.)

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, theselines converge from each vertex.

Group ExplorationGet into a group and experiment with the

different segments discussed.Every triangle has three examples of each

segment (three medians, three altitudes, etc.). Try constructing all three examples of a given segment and see what happens.

In your explorations, use a variety of triangles – acute, right, obtuse.

Triangle CentersHopefully you have observed that when you construct

all three medians, they intersect at a single point.This point is a center of the triangle.

When you construct all three altitudes they also intersect at a single point, but a different point from before! This is a different center.The same goes for angle bisectors and perpendicular

bisectors.A triangle has many different centers (see list). We will

study four:CircumcenterIncenterCentroidOrthocenter

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.

CircumcenterThe 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. (Proof)Circumcenters don’t have to be inside of the

triangle. (See p. 324 for examples.)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.

IncenterThe 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. (Proof)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.

CentroidA 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. (We can use a coordinate proof for this.)

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. (Let’s try this!)

OrthocenterIf 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 observationsIn 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 www.mathopenref.com.

Homework 22Workbook, pp. 59, 62

Handout