Math Journal 5

Perpendicular BisectorWhat is a Perpendicular Bisector? A Perpendicular Bisector a line or ray that passes through the midpoint of the segment

Theorem: If a line pass through a point in the segment that are equidistant from the endpoint then is a perpendicular bisector.

Converse: If the line that passes though a point in the segment and its not equidistant then is not a perpendicular bisector.

ExamplesA

BC D

1) If AB=25.4cm AD=20cm and DB=12cm find AB.

AB= 25.4

2) If H is perpendicular bisector of CB and DB is 4. Find CB.

CB= 8

3) Given that H is perpendicular bisector of segment CB and AB is 4x+4 and AC is 8x-6. Find AB

AB=AC 4x+6=8x-6-4x -4x

+6=4x-6+6 +6 4x=12X=3 AB=4(3)+4

AB=16

H

Angle Bisector

What is an Angle Bisector? A Ray or line that divides the angle is two congruent angle

Theorem: If a point is in the line of the angle bisector then is equidistant from the sides of the angles

Converse: If its equidistant from the sides of the angles then it on the angle bisector

Examples

A

B

C

D

1) <ABD congruent <ACD2) <DAC congruent <DAB3) <BDA congruent <CDA

Counter:1) Since AB is congruent to AB, AD is an angle bisector of triangle BAC2) Since BD is congruent to DC, AD is an angle bisector of triangle BAC

ConcurrentWhat is concurrent: Concurrent is when three or more lines meet together making a point.

the concurrency of Perpen-dicular bisectors of a triangle

theoremThe Perpendicular bisector of a triangle is a point where its equidistant from the vertices of the triangle

CircumcenterCircumcenter is where the point of the 3 per-pendicular bisector meets, its special because the point is equidistant through all the vertices of the triangle.

the concurrency of angle bisectors of a triangle

theoremWhen a point is made and is equidistant from the sides of the trian-gles.

IncenterA point made by the meeting of the three angle bisector, and it equidistant from the sides of the triangle.

INCEN-TER

MedianThe median of a triangle is a segment that goes from the vertex to the opposite side mid-segment

CentroidA point where all the median meets. The distant from the centroid to the vertex will the double from the distant from the centroid to the side.

the concurrency of medians of a triangle

theoremThe median of a triangle that in-tersect a point creating a line that is double the other part of the line

2

1

2

4

3

61.75

3.52.5

5

3.3

6.6

0.2

1.43

0.4

2.86

3.1

6.2

altitude of a triangleSegment that is perpendicular to the other angles.

OrthocenterThe point that all altitude lines meet of a triangle.

The concurrency of altitudes of a triangle theorem

A point that the altitudes inter-sects in a trian-gle.

Midsegment

What is Midsegment? Is a segment joined by the midpoint of the sides, the triangle has 3 midsegment in total.

Theorem: The midsegment is parallel to the opposite side and it’s half the size of the side.

the relationship between the longer and shorter sides of a triangle and

their opposite angles

In the triangle they are three segments that are related to the angle, meaning that if one segment is the biggest then to the opposite side an-gle will be the biggest one and if the segment is the shortest one the opposite side angle will be the shortest one.

*ATTENTION* This Images are not draw in scale :P

EX

AM

PLE

S

the exterior an-gle inequality

The angle that is an extension of a interior angle. A+B= the exterior angle

a

b

c

a

a

b

b

c

c

A+B

A+B

A+B

Triangle Inequality

The sum of 2 sides will be greater than the 3rd side.

1. 2,5,8 No, because adding 2+5<8 and this will not make a triangle.

2. 15,28,40 Yes because adding 15+28>40 and this will make a triangle

3. 16, 24, 40 No because adding 16+24=40 mak-ing two line with the same lengths.

Indirect Proof It’s a way to proof not directly but indirectly.

STEPS TO SOLVE THIS1)Assume the Given is false2) Use that as a give and start the proof3)When you come to a contradiction that will be your last step.

ExamplesProve A triangle cannot have 2 right angles

Statement Reason

Assume a triangle has 2 right <1=<2

Given

M<1=m<2=90 degrees Def of Right Angle

M<1+M<2=180 de-grees

Substitution

M<1+m<2+m<3=180 Triangle sum thrm

M<3=0 contradiction

A triangle cannot have 2 Right Angles

Statement Reason

Two supplementary an-gles can both be obtuse angles

Given

X>90 degrees Def of obtuse angles

(X>90)+(x>90) Substitution

(x>90)+(x>90)=180 contradiction

Proof: two supplementary angles cannot both be obtuse an-gles

Two supplementary angles cannot both be obtuse.

Statement Reason

A triangle can have 2 obtuse angles

Given

X>90 Def of obtuse angles

M<1+m<2+m<3=180 Def of a triangle

X can be 91 or more Def of obtuse angle

100+100+m<3=180 Substitution

200+M<3=180 Addition

M<3=-20 Subtraction

M<3=-20 contradiction

A Triangle cannot have 2 obtuse angles

Triangle cannot have 2 obtuse angles

Hinge TheoremTwo triangles with two congruent segments but the 3rd segment will different and will be larger if the angle that is in the opposite side will be bigger.

the special relationships in the special right triangles (30-60-90

and 45-45-90)

45-45-90: In this triangle the legs will the same size but the hypotenuse will be length times √2

30-60-90: In this triangle the base will the smallest size the longest side would be the hypotenuse that is the double of the base angle and the height will be the base time the √3

Examples

1

1

x X would be 1√2 because you multiply the legs time √2

16

X

X would be 8√2 because we need to divide 16 by √2, and we know that we can’t have the √2 as a denominator so we multiply ev-erything by √2 and later we have 16 √2 divided by 2 and we can simplify by 2 and the answer would be 8 √2

45

45

45

45

45

45

7x

X would be 7 √2/2

30

60

4

xy

X is going to be 8Y will be 4 √3

18

x

y

30

60

30

60

X= 9 because the hy-potenuse is the double of the base of the triangleY= is going to be 9 √3 be-cause we already know that the base is 9

9

x

y

X=3 √3Y=6 √3because to find the shortest side we need to dived 9 by √3 that is 3 √3 making X and we multiply time 2 the 3 √3

BOOM!!!!! FINISH