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Chapter 5 Review

Perpendicular Bisector, Angle Bisector, Median, Altitude,

Exterior Angles and Inequality

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Perpendicular Bisector – Special Segment of a triangle

AB PR

Definition: A line (or ray or segment) that is perpendicular to a segment at its midpoint.

The perpendicular bisector does not have to start from a vertex!

Example:

C D

In the scalene ∆CDE, is the perpendicular bisector.

In the right ∆MLN, is the perpendicular bisector.

In the isosceles ∆POQ, is the perpendicular bisector.

EA

B

M

L N

A B

RO Q

P

AB

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Median - Special Segment of Triangle

Definition: A segment from the vertex of the triangle to the midpoint of the opposite side.

Since there are three vertices, there are three medians.

In the figure C, E and F are the midpoints of the sides of the triangle.

, , .DC AF BE are the medians of the triangle

B

A DE

CF

P

Def: The centroid is the point of concurrency of the medians of a triangle.

x

2x y

2y

z 2z

NOTE: The distance from the vertex to the centroid is twice the distance from the centroid to the midpoint of the side opposite that vertex.

NOTE: The centroid is the center of balance (center of mass) or balance point for the triangle.

The centroid is ALWAYS inside the triangle.

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Altitude - Special Segment of Triangle

Definition: The perpendicular segment from a vertex of the triangle to the segment that contains the opposite side.

In a right triangle, two of the altitudes of are the legs of the triangle.

B

A DE

C

FB

A D

F

In an obtuse triangle, two of the altitudes are outside of the triangle.

, , .AF BE DC are the altitudes of the triangle

, ,AB AD AF altitudes of right B

A D

F

I

K , ,BI DK AF altitudes of obtuse

Where is the orthocenter with respect to each of the following triangles?

Acute Triangle

Obtuse Triangle

Right Triangle

Def: The point of concurrency of the three altitudes (or the lines containing the three altitudes) of a triangle is called

the orthocenter.

Inside the triangle

Outside the triangle

The vertex of the right angle

In each triangle do the altitudes intersect? The LINES containing the altitudes of triangle do intersect at one point.

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Exterior Angle Theorem

The measure of the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.

Exterior AngleRemote Interior Angles A

BC

D

m ACD m A m B

Example:

(3x-22)x80

B

A DC

Find the mA.

3x - 22 = x + 80

3x – x = 80 + 22

2x = 102

mA = x = 51°

NOTE: The median from the vertex angle of an isosceles triangle is also an angle bisector, a perpendicular bisector, and an altitude of the triangle.

The four points of concurrency all lie on this segment. (The points of concurrency of an isosceles triangle are collinear.)

What would be true about any median of an equilateral triangle???

Any median of an equilateral triangle is also an angle bisector, an altitude, and a perpendicular bisector.

NOTE: The centroid, incenter, orthocenter, and circumcenter are the same point in an equilateral triangle.

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Triangle Inequality – examples…

For the triangle, list the angles in order from least to greatest measure.

CA

B

4 cm

6 cm

5 cm

, ,

.

arg arg .

AB is the smallest side C smallest angle

BC is thel est side Ais

Angles in order from least to grea

the

tes

l est angle

t C B A

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Triangle Inequality – examples…

For the triangle, list the sides in order from shortest to longest measure.

8x-10

7x+67x+8

CA

B(7x + 8) ° + (7x + 6 ) ° + (8x – 10 ) ° = 180°

22 x + 4 = 180 °

22x = 176

X = 8

m<C = 7x + 8 = 64 °

m<A = 7x + 6 = 62 °

m<B = 8x – 10 = 54 °

64 °62 °

54 °

, ,

.

arg .

B is the smallest angle AC shortest side

C is thel est angle ABi

Sides in order from smallest to

s the

long

longest s

est AC BC AB

ide

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The perpendicular segment from a point to a line is the shortest segment from the point to the line.

Corollary 1:

The perpendicular segment from a point to a plane is the shortest segment from the point to the plane.

Corollary 2:

If one angle of a triangle is larger than a second angle, then the side opposite the first angle is larger than the side opposite the second angle.

Converse:

Converse Theorem & Corollaries

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Triangle Inequality Theorem:

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

c

b

a

B

C

A

a + b > c

a + c > b

b + c > a

Example: Determine if it is possible to draw a triangle with side measures 12, 11, and 17. 12 + 11 > 17 Yes

11 + 17 > 12 Yes

12 + 17 > 11 Yes

Therefore a triangle can be drawn.

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Finding the range of the third side:

Since the third side cannot be larger than the other two added together, we find the maximum value by adding the two sides.

Since the third side and the smallest side cannot be larger than the other side, we find the minimum value by subtracting the two sides.

Example: Given a triangle with sides of length 3 and 8, find the range of possible values for the third side.

The maximum value (if x is the largest side of the triangle) 3 + 8 < x

11 < x

The minimum value (if x is not that largest side of the ∆) 8 – 3 > x

5> x

Range of the third side is 5 < x < 11.