Chapter 5

More Triangles.

Mr. ThompsonMore Triangles.Mr. Thompson.

Midsegment Theorem

A midsegment of a triangle is a segment that connects the midpoints of

two sides of

a triangle.

Midsegment TheoremThe segment connecting the

midpoints of 2 sides of a triangle is parallel to

the 3rd side

and is ½ as

long.

Perpendiculars and Bisectors

In 1.5, you learned that a segment bisector intersects a segment at its midpoint.

10 10

midpoint

Segment bisector

A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is

called a perpendicular bisector.

12 12

d f

A point is equidistant from two points if its distance from each point is the same.

x z

y

Y is equidistant from X and Z.

Perpendicular Bisector Theorem If a point is on the perpendicular

bisector of a segment,

8 8A B

then it is equidistant from the endpoints of the segment.

Converse of the Perpendicular Bisector Theorem

If a point is equidistant from the endpoints of a segment,

x z

y

Y is equidistant from X and Z.

Using Perpendicular Bisectors

T

Q

S

M N

12

12

What segment lengths in the diagram are equal?

NS=NT (given)

M is on the perpendicular bisector of ST, so…..

MS=MT (Theorem 5.1)

QS =QT=12 (given)

Using Perpendicular Bisectors

T

Q

12

12

Explain why Q is on MN.

S

MN

QS=QT, so Q is equidistant from S and T.

By Theorem 5.2, Q is on the perpendicular bisector of ST, which is MN.

The distance from a point to a line….. defined as the length of the

perpendicular segment from the point to the line.

m

R

S

The distance from point R to line m is the length of RS.

Point that is equidistant from two lines…

When a point is the same distance from one line as it is from another line, the point is equidistant from the two lines(or rays or segments).

Angle Bisector TheoremIf a point is on the bisector of an angle,

then it is equidistant from the 2 sides of the angle.

If angle ABD = angle CBD,

then DC = AD.

Converse of the Angle Bisector TheoremIf a point is in the interior of an

angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle.

If DC = AD, then angle ABD = angle CBD.

Bisectors of a Triangle…

A perpendicular bisector of a triangle is a line (or ray or segment) that is perpendicular to a side of the triangle at the

midpoint of the

side.

Investigation…

…of the Perpendicular Bisector Theorem.

When three or more lines (or rays or segments) intersect in the same point, they are called concurrent lines (or rays or segments). The point of

intersection of the

lines is called the

point of

concurrency.

The three perpendicular bisectors of a triangle are concurrent. The point of concurrency can be inside the triangle, on the triangle, or outside the triangle.

The point of concurrency of the perpendicular bisectors

of a triangle is

called the

circumcenter

of the triangle.

Concurrency of Perpendicular Bisectors of a Triangle

The perpendicular bisectors of a triangle intersect at a point

that is equidistant

from the vertices

of the triangle.

OA1 = OA2 = OA3

An angle bisector of a triangle is a bisector of an angle of the triangle. The three angle bisectors are concurrent. The point of concurrency of the angle bisectors is called the

incenter of the triangle and is always inside the triangle.

Concurrency of Angle Bisectors of a TriangleThe angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. LMC = LMA = LMB

Classwork…

Page 246:6, 13, 31, 32, 35, 38

Page 252:28, 29, 33, 40, 46

Medians and Altitudes of a

Triangle

Medians and Altitudes

A median of a triangle is a segment whose endpoints are a vertex of the

triangle and the

midpoint of the

opposite side.

The three medians of a triangle are concurrent. The point of concurrency is called the centroid of the triangle. The centroid is always inside the triangle.

Concurrency of Medians of a Triangle

The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side.

An altitude of a triangle is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side. An altitude can lie inside, on or outside the triangle.

If AR, CT, and BU are altitudes of triangle ABC, then AR, CT,

and BU intersect at some point P.

Example1) ABC [A(-3,10),

B(9,2), and C(9,15)]:

a) Determine the coordinates of point P on AB so that CP is a median of ABC.

b) Determine if CP is an altitude of ABC

Example2) SGB [S(4,7), G(6,2),

and B(12,-1)]:

a) Determine the coordinates of point J on GB so that SJ is a median of SGB

b) Point M(8,3). Is GM an altitude of SGB ?

Inequalities in One Triangle

TheoremIf one side of a triangle is longer

than another side, then the angle opposite the longer side is larger than the angle

opposite the

shorter side.

TheoremIf one angle of a triangle is larger than

another angle, then the side opposite the larger angle is longer than the side opposite the smaller

side.

Triangle InequalityThe sum of the lengths of any two

sides of a triangle is greater than the length of the third side.

Would sides of length 4, 5 and 6 form a triangle....?

How about sides of length 4, 11, and 7 ?