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Chapter 5 Properties of Triangles

- The Bigger Picture
- Properties of Triangles such as perpendicular and

angle bisectors and how they relate in triangles - Congruency of perpendicular and angle bisectors
- - The use of medians and altitudes to locate

points on a triangle

The What and the Why

- Use properties of mid-segments of a triangle
- - Determining length
- Compare the length of sides or the measures of

the angles of a triangle - Determining balance points for structural

projects such as cranes and booms - Understand and write indirect proofs
- Prove theorems that cannot be easily proved

directly - Use the Hinge Theorem and its converse to compare

side lengths and angle measures of triangles. - - Comparative analysis tool for measuring

distance to a common point

- Use of properties of perpendicular bisectors and

angle bisectors - Optimize the positioning of a goalie in hockey or

soccer - Use properties of perpendicular bisectors and

angle bisectors - - Using the concept of equidistant to locate

objects and points - Use properties of medians and altitudes of a

triangle - - Find points on a triangle and use them to

measure various objects such as a persons heart

fitness

Properties of Triangles

- Soccer goalkeepers use triangle relationships to

help block shots on goal. - An opponent can shoot the ball from many

different angles. The goalkeeper determines the

best defensive position by imagining a triangle

formed by the goal posts and the opponent. - The opponent x is trying to score a goal. Which

position do you think is best for the goalkeeper,

A, B, or C? Why? - Estimate the measure of ltX, known as the shooting

angle. How could the opponent change positions

to improve the shooting angle?

Using Properties of Perpendicular Bisectors

In chapter 1 we learned that a segment bisector

intersects a segment at its midpoint. A segment,

ray, line, or plane that is perpendicular to a

segment at its midpoint is called a

Perpendicular Bisector.

CP is a __ bisector of AB

Using Perpendicular Bisectors

Using the Properties of Angle Bisectors

Consider the following the distance from a

point to a line is defined as the length of the

perpendicular segment from the point to the line.

And, when a point is the same distance from

one line as it is from another, then the point is

equidistant from the two lines. Knowing that,

we can then apply the logic to a point on the

interior of an angle, and use it to help

determine the bisector of the angle.

Angle Bisector Theorems

Theorem 5.3 Angle Bisector Theorem If a point is

on the bisector of an angle, then the point is

equidistant from the two sides of the angle. If

mltBAD mltCAD, then DB DC Theorem 5.4

Converse of Angle Bisector Theorem If a point is

in the interior of an angle and is equidistant

from the sides of the angle, the it lies on the

bisector of the angle. If DB DC, then mltBAD

mltCAD

Proof of Theorem 5.3 Angle Bisector Theorem

Bisectors of a Triangle

Perpendicular Bisector of a Triangle A line

(ray of segment) that is perpendicular to the

side of the triangle at its midpoint. Concurrent

Lines Three of more lines that intersect at the

same point. Point of Concurrency The point of

intersection of the lines is called the point of

concurrency. The three perpendicular bisectors

of a triangle are concurrent. And the point of

concurrency is known as the Circumcenter and can

be located inside, on, or outside the triangle.

Theorem 5.5 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. PA PB PC

The point of concurrency (P) of the perpendicular

bisectors of the triangle is called the

Circumcenter of the Triangle Where and how might

this be helpful in the real world?

Perpendicular Bisectors of a Triangle and

Facilities Planning an example

Client F

Client E

Client G

A company is planning to build a new distribution

facility that is convenient to all of its major

clients. How might locating the circumcenter of

the three clients be beneficial in determining

the location?

Using Angle Bisectors of a Triangle

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 is called the incenter of the

triangle. Theorem 5.6 Concurrency of Angle

Bisectors of a Triangle The angle bisectors of a

triangle intersect at a point that is equidistant

from the sides of the triangle. PD PE PF

Using Angle Bisectors

Based upon the Angle Bisector Theorem, which

segments are congruent?

If the length of ML 17, and the length of MQ

15, can we determine the length of LQ, LS, and LR?

Medians and Altitudes of Triangles

Median a segment whose endpoints are a vertex

and the midpoint of the opposite side of the

triangle. The three Medians are Concurrent The

point of concurrency is called the Centroid of

the triangle

Theorem 5.7 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. If P is the centroid of /\ ABC, then, AP

2/3 AD, BP 2/3 BF, and CP 2/3CE

Altitude of a Triangle

An altitude of a triangle is a perpendicular

segment from a vertex to the opposite side of a

triangle, or to a line that contains the opposite

side of the triangle. An altitude can lie

inside, outside, or on the triangle. Altitudes

are concurrent, and the point of concurrency is

called the orthocenter of the triangle.

Summary Triangle Bisectors

Perpendicular Bisector of a Triangle A line

(ray of segment) that is perpendicular to the

side of the triangle at its midpoint. The point

of concurrency is called the Circumcenter.

An Angle Bisector of a triangle is a bisector of

an angle of the triangle. The point of

.concurrency is called the Incenter.

An altitude of a triangle is a perpendicular

segment from a vertex to the opposite side of a

triangle, or to a line that contains the opposite

side of the triangle. An altitude can lie

inside, outside, or on the triangle. Altitudes

are concurrent, and the point of concurrency is

called the orthocenter of the triangle.

Median a segment whose endpoints are a vertex

and the midpoint of the opposite side of the

triangle. The three Medians are Concurrent and

the point of concurrency is called the Centroid

of the triangle

Mid-Segment Theorem

A Mid-segment of a triangle is a segment that

connects the midpoints of two sides of a triangle.

Theorem 5.9 Mid-segment Theorem The segment

connecting the midpoints of two sides of a

triangle is parallel to the third side, and is

half as long. DE AB and DE ½ AB

Using the Mid-segment Theorem

UW and VW are mid-segments of Triangle RST. VW

8, RS 12. Find UW and RT.

Proving the Mid-segment Theorem

1. DE is the mid-segment for sides AC and BC. 2.

Using the midpoint formula, we can determine the

coordinate values for points D and E. D (5,

5) E (11, 5) 3. Determine the slope of DE

- Compare that to the slope of AB 4.

Determine the length of DE and AB using the

distance formula. - Since they are both

horizontal lines, the length can be determined as

the absolute value of the difference in the x

values. DE 6 AB 12 5. DE AB DE

½ AB

C(10, 10)

D E

A(0, 0)

B (12, 0)

Using Midpoints to Draw a Triangle

The midpoints of the sides of a triangle are L

(4, 2), M(2, 3), and N(5, 4). What are the

coordinates of the vertices of the triangles?

Perimeter of a Mid-segment Triangle

Given ST 12, TR 10, and SR 8, What is the

perimeter of Triangle UVW?

Hinge Theorem

Theorem 5.14 Hinge Theorem If two sides of one

triangle are congruent to two sides of another

triangle, and the included angle of the first is

larger than the included angle of the second,

then the third side of the first triangle is

longer than the third side of the

second. Theorem 5.15 Converse of the Hinge

Theorem If two sides of one triangle are

congruent to two sides of another triangle, and

the third side of the first is longer than the

third side of the second, then the included angle

of the first is larger than the included angle of

the second.

Indirect Proof of Theorem 5.15 Example 2

Finding Possible Side Lengths and Angle Measures

E

B 80

36 D

F

A C

Comparing Distances using the Hinge Theorem

You and a friend are flying separate planes. You

leave the airport and fly 120 miles due west.

You then change direction and fly W 30 N for 70

miles. Your friend leaves the airport and flies

120 miles due east. She then changes direction

and flies E40 S for 70 miles. Each of you have

flown exactly 190 miles, but which one of you is

farther from the airport?

P airport

Comparing Distances using the Hinge Theorem

- Your flight 100 miles due west, then 50 miles

N20 W. - Your Friend 100 Miles due north, then 50 miles

N30 E - 2. Your flight 210 miles due south, then 80

miles S70 W. - Your Friend 80 miles due north, then 210

miles N50 E.

P airport

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