1. Day 36 Triangle Segments and Centers
Geometry
Day 36
Triangle Segments and Centers

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

3. 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.
Proof: C A B D

4. 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.
Write this proof in groups. Given: AC  BC, AD  BD Prove: C and D are on the perpendicular bisector of AB
Hint: Use a larger pair of congruent triangles to prove that a smaller pair of triangles are congruent. C A B D

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

6. Perpendicular bisector
See the following video for constructing a perpendicular bisector.

7. Angle Bisector
Remember 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).

8. 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!
Proof: C A B D

9. 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.
Write the proof as a group. Given: AD = BD; AD  AC; BD  BC Prove: CD bisects ACB C A B D

10. 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.

11. 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).
Why is this?

12. 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).

13. 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.

14. Altitude Complete the handout on altitudes.
To construct an altitude, we need to know how to create perpendicular lines. (See video.)

15. 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.

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

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

18. Group Exploration
Get 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.

19. Triangle Centers
Hopefully 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:
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.

20. 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. (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.

21. 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. (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.

22. 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. (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!)

23. 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.

24. 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

25. Homework 22
Workbook, pp. 59, 62 Handout