§ 6.1 Medians Medians § 6.4 Isosceles Triangles Isosceles TrianglesIsosceles Triangles § 6.3...

More About TrianglesMore About TrianglesMore About TrianglesMore About Triangles

§§ 6.1 6.1 Medians

§§ 6.4 6.4 Isosceles Triangles

§§ 6.3 6.3 Angle Bisectors of Triangles

§§ 6.2 6.2 Altitudes and Perpendicular Bisectors

§§ 6.6 6.6 The Pythagorean Theorem

§§ 6.5 6.5 Right Triangles

§§ 6.7 6.7 Distance on the Coordinate Plane

MediansMedians

You will learn to identify and construct medians in triangles

1) ______

2) _______

3) _________

median

centroid

concurrent

MediansMedians

In a triangle, a median is a segment that joins a ______ of the triangle andthe ________ of the side __________________.

vertexmidpoint opposite that vertex

C

B

A

D

EF

BEmedian

ADmedian

CFmedian

centroidThe medians of ΔABC, AD, BE, and CF, intersect at a common pointcalled the ________.

When three or more lines or segments meet at the same point, the lines are__________. concurrent

MediansMedians

There is a special relationship between the length of the segment from thevertex to the centroid

DC

B

A

E

F

and the length of the segment from the centroid to themidpoint.

MediansMedians

Theorem 6 - 1

The length of the segment from the vertex to the centroid is

_____ the length of the segment from the centroid to the midpoint.twice

x

2x

When three or more lines or segments meet at the same point, the lines are __________.concurrent

MediansMedians

DC

B

A

E

F

. of medians are and , , ABCCFBEAD

?92 and ,15

,34 CE if of measure theisWhat

xEAxDB

xCD

CD = 14

Altitudes and Perpendicular BisectorsAltitudes and Perpendicular Bisectors

You will learn to identify and construct _______ and__________________ in triangles.

1) ______

2) __________________

altitudesperpendicular bisectors

altitude

perpendicular bisector


In geometry, an altitude of a triangle is a ____________ segment with oneendpoint at a ______ and the other endpoint on the side _______ thatvertex.

perpendicularvertex opposite

D

The altitude AD is perpendicular to side BC.

C

A

B


CA

B

Constructing an altitude of a triangle

1) Draw a triangle like ΔABC2) Place the compass point on B and draw an arc that intersects side AC in two points. Label the points of intersection D and E.

3) Place the compass point at D and draw an arc below AC. Using the same compass setting, place the compass point on E and draw an arc to intersect the one drawn.

4) Use a straightedge to align the vertex B and the point where the two arcs intersect. Draw a segment from the vertex B to side AC. Label the point of intersection F.

D E


An altitude of a triangle may not always lie inside the triangle.

Altitudes of Triangles

acute triangle right triangle obtuse triangle

The altitude is_____ the triangle

The altitude is _____ of the triangle

The altitude is _______ the triangleinside out sidea side


Another special line in a triangle is a perpendicular bisector.

A perpendicular line or segment that bisects a ____ of a triangle is called theperpendicular bisector of that side.

side

A

B CD

D is the midpoint of BC.

m altitude

Line m is the perpendicular bisectorof side BC.


In some triangles, the perpendicular bisector and the altitude are the same.

X Z

Y

E

YE is an altitude.

The line containing YE is theperpendicular bisector of XZ.

E is the midpoint of XZ.

Angle Bisectors of TrianglesAngle Bisectors of Triangles

You will learn to identify and use ____________ in triangles.

1) ___________

angle bisectors

angle bisector

Angle Bisectors of Triangles Angle Bisectors of Triangles

Recall that the bisector of an angle is a ray that separates the angle into twocongruent angles.

S

Q

R

P

SPRmQPSm

SPRQPS

QPRPS

bisects


An angle bisector of a triangle is a segment that separates an angle of the triangle into two congruent angles.

One of the endpoints of an angle bisector is a ______ of the triangle,

and the other endpoint is on the side ________ that vertex.

vertex

opposite

A

B

D

C

CABmDACm

CABDAC

DAB

ofbisector anglean is AC

Just as every triangle has three medians, three altitudes, and three perpendicular bisectors, every triangle has three angle bisectors.


Special Segments in Triangles

Segment

Type

Property

altitude perpendicular bisector

angle bisector

line segment line ray

line segment line segment

from the vertex, aline perpendicularto the opposite side

bisects the sideof the triangle

bisects the angleof the triangle

Isosceles Triangles Isosceles Triangles

You will learn to identify and use properties of _______triangles.

1) _____________

2) ____

3) ____

isosceles

isosceles triangle

base

legs


Recall from §5-1 that an isosceles triangle has at least two congruent sides.

The congruent sides are called ____.legs

The side opposite the vertex angle is called the ____.base

In an isosceles triangle, there are two base angles, the vertices where thebase intersects the congruent sides.

vertex angle

leg leg

basebase angle base angle


Theorem

6-2

Isosceles

Triangle

Theorem

6-3

If two sides of a

triangle are congruent,

then the angles

opposite those sides

are congruent.

The median from thevertex angle of an isosceles triangle lieson the perpendicularbisector of the baseand the angle bisectorof the vertex angle.

A

B C

BC

ACAB

then, If

CADBAD

BCAD

CD

ACAB

and

then,BD

and IfA

B CD


Theorem

6-4

Converse of

Isosceles

Triangle

Theorem

If two angles of a

triangle are congruent,

then the sides opposite those angles

are congruent.

A

B C

ACAB

CB

then, If

Theorem

6-5

A triangle is equilateral if and only if it is equiangular.

Right Triangles Right Triangles

You will learn to use tests for _________ of ____ triangles.

1) _________

2) ____

congruence right

hypotenuse

legs


In a right triangle, the side opposite the right angle is called the_________.hypotenuse

hypotenuse

The two sides that form the right angle are called the ____.legs

leg

leg


Recall from Chapter 5, we studied various ways to prove triangles to becongruent:

In §5-5, we studied two theorems

and

A

B

C R

S

T

A

B

C R

S

T


Recall from Chapter 5, we studied various ways to prove triangles to becongruent:

In §5-6, we studied two theorems

and

R

S

TA

B

C

R

S

TA

B

C


The theorems mentioned in Chapter 5, were for ALL triangles.So, it should make perfect sense that they would apply to right trianglesas well.

Theorem

6-6

LL Theorem

If two legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent.

A

CB

D

FE

same as

DEFABC



Theorem

6-7

HA Theorem

If ______________ and an (either) __________ of one right triangle are congruent to the __________ and

_________________ of another right angle, then the

triangles are congruent.

same as

DEFABC

the hypotenuse acute anglehypotenuse

corresponding angle

A

CB

D

FE



Theorem

6-6

LA Theorem

If one (either) ___ and an __________ of a right triangle are

congruent to the ________________________ of another right triangle, then the triangles are congruent.

same as

DEFABC

leg acute anglecorresponding leg and angle

A

CB

D

FE



Postulate

6-1

HL Postulate

If the hypotenuse and a leg on one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent.

A

CB

D

FE

DEFABC Theorem?

Pythagorean Theorem Pythagorean Theorem

You will learn to use the __________ Theorem and its converse.

1) _________________

2) _______________

* 3) _______

Pythagorean

Pythagorean Theorem

Pythagorean triple

converse


If ___ measures of the sides of a _____ triangle are known, the___________________ can be used to find the measure of the third ____.

two rightPythagorean Theorem side

a

b

c 222 bac

22 bac

A _________________ is a group of three whole numbers that satisfiesthe equation c2 = a2 + b2, where c is the measure of the hypotenuse.

Pythagorean triple

35

4

52 = 32 + 42

25 = 9 + 16


Theorem

6-9

Pythagorean Theorem

In a right triangle, the square of the length of the hypotenuse __, is equal to the sum of the squares of the

lengths of the legs __ and __.

c

b

a

cba

222 bac

Theorem 6-10

Converse of the

Pythagorean Theorem

If c is the measure of the longest side of a triangle, a and b are the lengths of the other two sides, and c2 = a2 + b2,

then the triangle is a right angle.

Distance on the Coordinate Plane Distance on the Coordinate Plane

You will learn to find the ______________________on thecoordinate plane.

Nothing new! You learned this in Algebra I.

212

212 yyxxd

distance between two points


1) On grid paper, graph A(-3, 1) and C(2, 3).

y

x

2) Draw a horizontal segment from A and a vertical segment from C.

A(-3, 1)

C(2, 3)

B(2, 1)

3) Label the intersection B and find the coordinates of B.

QUESTIONS:

What is the measure of the distance between A and B?

What is the measure of the distance between B and C?

What kind of triangle is ΔABC?

If AB and BC are known, what theorem can be used to find AC?

What is the measure of AC?

(x2 – x1) = 5

(y2 – y1) = 2

right triangle

Pythagorean Theorem

29 ≈ 5.4


Theorem

6-11

DistanceFormula

If d is the measure of the distance between two points with

coordinates (x1, y1) and (x2, y2),

212

212 yyxx

y

xA(x1, y1)

B(x2, y2)

d

then d =

Find the distance between each pair of points.Round to the nearest tenth, if necessary.

7,5,3,2 N M 5

2,2,4,6 U T 4.5


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