PROPERTIES OF TRIANGLES

Chapter 5

Vocab Review Intersect Midpoint Angle Bisector Perpendicular Bisector

Construction of a Perpendicular through a point on a line

Perpendicular Bisector Theorem

If a point is on the perpendicular bisector of a segment,

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 the segment ,

then it is on the

perpendicular bisector of a segment

Using Perpendicular Bisectors

What segment lengths are equal?

Why is L on OQ?

Given: OQ is the bisector of MP

Angle Bisector Theorem

If a point is on the bisector of an angle,

then it is

equidistant from the two side of the angle

Converse of the Angle Bisector Theorem

If 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

Angle Bisector Theorem Let’s Prove

it…

Given: • D is on the

bisector of BAC• BD AB• CD AC

Prove: DB DC

STATEMENTS

REASONS

“Guided Practice”Page 267 #1-7

CHECK POINT

Perpendicular Bisector of a Triangle

Line (or ray/ segments) that is perpendicular to a side of the triangle at the midpoint of the side.

Concurrent @ Circumcenter

(not necessarily inside the triangle)

Perpendicular Bisector of a Triangle

Concurrency

Point of Concurrency The point of

intersection of the lines

Concurrent Lines When 3 or more

lines (rays/segments) intersect at the same point. .

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 AG = BG =

CG

Circumcenter

Exit Ticket

Angle Bisector of a Triangle

Bisector of an angle of the triangle.(one for each angle)

Concurrent @ Incenter (always inside

the triangle)

Angle Bisector of a Triangle

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 EG = DG =

FG

Incenter

Pythagorean Theorem

a2 + b2 = c2

Pythagorean Theorem

a2 + b2 = c2

EXAMPLE:a= ?, b= 15, c=

17a2 + b2 = c2

a2 + 152 = 172

a2 + 225 = 289a2 = 64a = 8

ClassworkPage 275-278

#10-23,

CHECK POINT

Median of a Triangle

Segment whose end points are a vertex of the triangle and the midpoint of the opposite side.

Concurrent @ Centroid(always inside

the triangle)

Median of a Triangle

Concurrency of Medians of a Triangle

The medians of a triangle intersect at a point

that is ²/₃ of the distance from each vertex to the midpoint of the opposite side.AG = ²/₃AD, BG = ²/₃BE, CG

= ²/₃CF

Centoid

Page 280 Example #1

EXAMPLE

Altitude of a Triangle

Perpendicular segment from a vertex to the opposite side.

Concurrent @ Orthocenter(inside, on, or outside the

triangle)

Page 281 Example #3

EXAMPLE

Concurrency of Altitudes of a Triangle

Lines containing the altitudes of a triangle are concurrent

Intersect at some point (H) called the orthocenter

(Inside, On, or Outside)

orthocenter

Activity

Video

Classwork:pg 282 #13-16

Homework:pg 282 #1-11, 17-20, &

34

CHECK POINT

Warm-upFind the coordinates of the midpoint:

1. (2, 0) and (0, 2)2. (5, 8) and (-3, -4)

Find the slope of the line through the given points:

1. (3, 11) and (3, 4)2. (-2, -6) and (4, -9)3. (4, 3) and (8, 3)

Midsegment of a Triangle

*Segment that connects the midpoints of two sides of a triangle

*Half as long as the side of the triangle it is parallel to

** REVIEW **

MIDPOINT FORMULA

Midpoint =

Midsegment of a Triangle

Find a Midsegment of the triangle….

Let D be the midpoint of AB

Let E be the midpoint of BC

Midsegment Theorem

The segment connecting the midpoint of two sides of a triangle is parallel to the third side and is half as long

DEBC and DE=¹₂BC⁄

Prove it…

How do you prove DEBC?

How do you prove DE=¹₂BC?⁄

Use the slope formula to find each slope (parallel lines have the same slope)

Use the distance formula to find the distance of each segment

Exit TicketPg 290 #3-9

Classwork:pg 290 #10-18, 21-25

CHECK POINT

Warm-upIn ∆DEF, G, H, and J are the midpoints of the sides…

J

H

G

F

ED1. Which segment is parallel to EF?

2. If GJ=3, what is HE?3. If DF=5, what is HJ?

GH

3

2.5

Pgs 310-311#1-15

Pgs 313#1-11

Test Review