JOURNAL CHAPTER 5

By: Ana Julia Rogozinski (YOLO)

Perpendicular bisector- A perpendicular bisector is the division of a line when making two congruent halves by passing through its midpoint, creating right angles.-EXPLANATION: When you have a segment and you want to construct a perpendicular bisector your first step would be to find the midpoint of the segment, by using a compass. Then you draw a line right between the middle and remember it has to be perpendicular to the line meaning both sides will be the same and will form angles of 90° (right angles).

Examples:

A B

W

Y

E

1. YA = YB

2.WA=WB

3.EA= EB

Theorem and itsconverse

-Perpendicular bisector theorem states: The Perpendicular bisectors of the sides of a triangle are concurrent at a point that is equidistant from the vertices of the triangle--If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.-Converse: Is a point is equidistant from the endpoints of a segment, then it is in the perpendicular bisector of the segment.

Examples:A

B CZ

1. CZ= ZB

2. AB= AC

XA

N

3. What can you prove by the converse of the perpendicular bisector theorem? You can prove XA= AN

Angle bisector-an angle bisector, it is a line passing through the vertex of the angle that cuts it into two equal smaller angles (in half). The bisector is what is cutting the angle or triangle in half.

Examples:

Y

WX

F2x+5

Example 3: If X =2 then what is the measurement of YX? 9

Theorem and its converse-Angle bisector theorem: If there is a point bisecting an angle then it is equidistant from the sides of the angle. -Converse: If the point inferior of an angle is equidistant from the sides of the angle, then is is on the bisector of the angle.

EXAMPLES:

Concurrency-- Concurrent means the three or more lines that intersect all at a point.- The point of concurrency is the point in which all lines intersect. - The concurrency of perpendicular bisector of a triangle is when the three sides of the triangle all have a perpendicular line bisecting them in the middle and when those lines intersect they are said to be concurrent. It’s theorem states that the circumcenter of a triangle is equidistant from the vertices of the triangle.

Examples of concurrent:Example 1 Example 2

Example 3

Examples of perpendicularbisector of a triangle

Example 1Example 2

Example 3

Circumcenter-Circumcenter is the point where the three perpendicular bisectors intersect. The circumcenter can lie on the inside of the triangle and sometimes even outside of it. Its theorem states that: a triangle is equidistant from the vertices of the triangle.

EXAMPLES OF CIRCUMCENTER

Example 1Example 2

Example 3

An angle bisector of a triangle is when the three angles of a triangle are bisected creating 2 equal angles for each and when the bisecting lines that intersect are concurrent, that point is the incenter of a triangle.

It’s theorem states that the incenter of a triangle is equidistant from the sides of the triangle.

- An incenter is the center of a triangles inscribed circle. -An incenter is equally distant from the sides of the triangle.

Incenter

Examples of incenterExample 1

Example 2

Example 3

MediansA median is a line in a triangle in which one end is in the middle of one of its side and the other is the vertex. Every triangle has three medians and when they intersect they are concurrent. -Medians intersect at the centroid.

Centroid-Theorem: The centroid of a triangle is 2/3 of the distance from each vertex to the midpoint of the opposite side

EXAMPLES OF MEDIANS

Examples of centroidExample 1 Example

2

Example 3

Concurrency of Medians of a

Triangle Theorem

Altitude of a triangle- Altitudes intersect at the orthocenter.- The altitude of a triangle is a line segment from one vertex of a triangle to the opposite side so that the line segment is PERPENDICULAR to the side.-Some altitudes can even fall outside the triangle. For example in obtuse triangle.Orthocenter

The point where the three altitudes of a triangle intersect.

Examples of altitudesExample 1 Example 2

Example 3

Examples of OrthocenterExample 1 Example 2

Example 3

Concurrency of Altitudes of triangles

Midsegments-- A midsegment is a segment that connects the midpoints of two sides of a triangle; The midsegment triangle is created since every triangle has 3 midsegments. Three midsegments intersect to form a triangle with ½ the perimeter of the larger triangle

-Theorem: A midsegment connecting two sides of a triangle is parallel to the third side and is half as long

Examples:Example 1

Example 2

X= 3 because PQ = ½ of BC

Example 3

If AD=DB and AE = EC, then DE || BC

The longer and shorter Sides of a triangle and their

opposites-When having only some measurements of the sides and of some angles, in order for you to know all the sides or angles in the triangle from the biggest to smaller, you can use this. - When having the angles and you want to see the order of the sides for example, you would see the opposite angle from each side and depending on the measurement of the angles , you will see the measurement of the sides. For example the bigger angle on a triangle ,its opposite side will be the bigger as well.

ExamplesExample 1

3230

38

What are the order of the angles from the smallest to the biggest?F, g, ee

fgExample 2

78 62

40What is the order of the sides from bigger to smaller?U, M , R

R

UM

Example 3

11.2 11

11.3

N

F P

Which angle is the smalles? F

Examples:Example 1

Which is the longer side?CF

C

F W

35

54 61

Example 2: Which is the shortest side? FW

Example 3: Order the sides from greatest to shortest: CF , CW, FW

EXTERIOR ANGLE INEQUIALITY

-Since we know from the exterior angle theorem that the exterior angle is the sum of the 2 other interior angles , we know then that the exterior angle must be larger that either of the angles that need to be added.-An exterior angle of a triangle is greater than either of the non-adjacent interior angles.

Examples:1. Which angle is greater

<ADX or < XDR? <ADX2. Which angle is greater

<DXR or <TXD? < TXD3. <QRD id greater that < DRX

A

D

XT R Q

Triangle inequality-The triangle inequality states that for any triangle the sum of two of the lengths of any two sides must be greater than the length of the remaining side.

Examples:1. Can 2.23, 2.12 ,4 make a triangle?

YES, because the sum of 2.23 and 2.12 is greater than 4.

2. Can 3.5, 4.8, 8 make a triangle?NO, because they do not add up to more than 8

3. What type of triangle do 9,14,17 create? They create an obtuse triangle

Indirect proofs-Indirect proofs are used when it is not possible to prove something directly.-Steps to write an indirect proof: 1st: State all possibilities2nd:Assume that what you are proving is FALSE. 3rd: Use that as your given, and start proving4th: When you come to a contradiction you have proved that is true.

Examples:

Statement Reason1. Assume a triangle has 2 right <‘s2.m<1= m<2=903.m<1+m<2=1804.m<1+m<2+m<3 = 1805. m<3= 06.----------------

1. Given

2. Definition right <

3. Substitution4. Triangle sum

theorem

5. Contradiction6. So a triangle

cannot have 2 right angles.

Example 1: Prove that a triangle cannot have 2 right <‘s

Example 2: Prove that If a > 0 then 1/a > 0

1. Assume a>0 so 1/a>0

2. 1/a > 0 ∙a ∙a3. 1<0

Statement

Reason

1. Given

2. Multiplicative prop.

3. Contradiction, so a>0 then 1/a>0 is false.

Example 3

Statement Reason1.< BDA is a straight angle2.m<BDA=1803. AD is perpendicular to BC4. <BDA is a right angle5. m<BDA= 906. m<BDA is not a straight angle7. BDA is not a straight angle

1. Assume opposite2. Definition of a

straight angle3. Given4. Def. of

perpendicular lines

5. Def. of a right <6. Contradiction7. We know by the

contradiction that its not a straight angle

Prove that  BDA is not a straight angle

Hinge Theorem-The hinge theorem states that If two triangles have two sides that are congruent, but the third side is not congruent, then the triangle with the larger included angle has the longer third side.-Converse: If the triangle with larger included angle has the longer third side, then the two triangles have two sides that are congruent but the third side is not congruent.

ExamplesExample 1

B

A

C Y Z

X

What can you prove by the hinge theorem?

-You can prove that these triangles are the same because the angles are equal/

Example 2

KZ

A

WL

B

Which side has a greater measurement?Line KW is longer because the angle is bigger.

42°

37°

Example 3

62°50°

JI

H

G

Complete using <, > or = JG < Ji

Special right Triangles-In 30-60-90 triangles the hypotenous is 2 times of the shorter led, and the length of the longer leg is the length of the shorter leg times √3-In 45-45-90 triangles both legs are congruent , 3 the length of the hypotenoues is the lengt of a leg times √2.

45°45°90° Examples

16X

X

Example 1 16 16√2 =8√2√2 2

Example 2

3

3

X

3² + 3²=x² 9 + 9 =x² √18 = √x² x= 3 √2

Example 3

X

9

X= 9√2 2

30°60°90° ExamplesExample 1 Example 2

22x

y

60°

30°

X= 11Y= 11√3

X

Y

18√3X=9√3Y= 27

Example 3

10√3

XY

X= 10Y= 20

THE END :P