TRIANGLES!!

Today, you will learn about various characteristics and properties of triangles … so

get your pencils ready because there are A LOT of

definitions!

Bisectors of a Triangle

Perpendicular bisector of a triangle: a line/ray/segment that is perpendicular to a side of the triangle at the midpoint of the side.

Perpendicular Bisectors of different triangles

Concurrency

When three or more lines intersect in the same point, they are called concurrent lines.

The point of intersection of the lines is called the point of concurrency.

Thm:Concurrency of Perpendicular Bisectors of a

TriangleThe perpendicular

bisectors of a triangle intersect at a point that is equidistant from the vertices of a the triangle.

*perpendicular bisectors*distance from circumcenter to

vertex (all congruent)

The point of concurrency of the perpendicular bisectors of a triangle

is called the circumcenter.

According to our thm., AP = BP = CP.

(Circumcenter is equidistant from vertices)

A

B

C

P

Bisectors of a Triangle

Angle bisector of a triangle: a line/ray/segment that bisects the angle of a triangle.

Thm:Concurrency of Angle Bisectors of a Triangle

The angle bisectors of a triangle intersect at a point that is equidistant from the sides of a the triangle.

*angle bisectors

*distance from incenter to sides (all congruent)

The point of concurrency of angle bisectors of a triangle is called the

incenter.

According to our thm., KM = JM = ML.

(Incenter is equidistant from sides)

J

L

MK

Example

a. Assuming there is a point at A,B, and C, which point is the best spot to place a sprinkler to water plants located at X,Y, and Z?

b. If the sprinkler covers a circular region with a radius of 15 in., will water reach all three plants?

Z

XY

C

B

A

10 in.

Medians of a triangle

Median of a triangle: a segment that starts at the vertex of a triangle and ends at the midpoint of the opposite side.

Thm: Concurrency of Medians of a Triangle

The medians of a triangle intersect at a point such that the distance from the vertex to that point is two-thirds of the length of the entire median within the triangle.

* Medians of the triangle

The point of concurrency of medians of a triangle is called the centroid.

According to our thm., PM = (2/3)PB. (This holds for all 3 medians.)

P

Q

R

B

C

A

M

Altitudes of a Triangle

Altitude of a triange: a line that starts at the vertex and ends at the opposite side (OR line that contains the opposite side) such that is is perpendicular to the opposite side.

Altitudes of different triangles

Concurrency of Altitudes of a Triangle

The altitudes of a triangle intersect at a point called the orthocenter.

* Altitudes of the triangle

Example

C is the centroid of GHJ and CM = 8.

a. Find HM

b. Find CH

H

JG

C

Example

Where is the orthocenter located in ABC if:

a. m of angle A = m of angle B = m of angle C

b. m of angle A = m of angle B = 45 degrees

c. m of angle A = 110 degrees

Checkpoint!!

1. What is an altitude?

2. What is the pt of concurrency of altitudes?

3. What is a perpendicular bisector?

4. What is the pt of concurrency of per. bis.?

5. What is a median?

6. What is the pt of concurrency of medians?

7. What is an angle bisector?

8. What is the pt of concurrency of angle bisectors?