1) what slides did you change? - Wikispaces.pdf/... · right and

Absent October 19, 2013

1) what slides did you change?2) based on data.what quizzes did you make up w keys3)based on data what slides and scalfold did you add?4)DID YOU READ LESSON!! what are you teaching?-------------------------------------------spiral and test key..break down!! this will help you tooon what to add!!!-------------------------------------------

notes midsegment gg 42 43 l23midpoint formula gg 66 67incenter, circumcenter, orthocenter, centroidtriangle inequality g33


1) Find the slope of the line that passes through the points (5,-2) and (7,-8)

2) What is the slope of the line 3x - 5 = 2y?

3) What are some ways to prove that two lines are parallel?

AIM: How do we find the midsegment of a triangle? G.G.42, G.G.43, L23 in CoachDO NOW:


quiz..and boost grade

interactmath.com on hw

regent handout



graph paper..on regents..

how could you do it that way?


Your test should have been marked like this:

You should TRACE the triangles when you take the test so you realize which ones they are talking about!!!

S: PR is congruent to SQA: PR RQ and SQ RQ means <PRQ is right and <SQR is right - - they are therefore CONGRUENT!S: RQ is conguruent to itself by reflexive property


Solve in your notebook:

Solution:11 = 2x - 516 = 2x8 = x

Hint:If the two baseangles are congruentwhat's true about the two legs?

what type of triangle is it?

why is this key??


m

n

k

P

DRAW IT FIRST!!!!!!

a


If you can't visualize it in your head - DRAW IT.

F

G

H42 104

Solution<F + <G = 10442 + <G = 104<G = 104 - 42<G = 62

2

asher print..(then take me this out)


Midsegment of a Triangle

Definition: segment that connects the midpoints of two sides of a triangle

Properties: parallel to 3rd side and 1/2 it's length

A

c

D

B

e

10

5



MODEL


video on it

1.3 minutes


what else is equal

ae= eb ad= dc


DE is parallel to AC

DE = (1/2)AC




model...copy


Video


• UW = ½(RS) = ½ (12) = 6 • RT = 2(VW) = 2(8) = 16

uw ll rs

Use the drawing and the given information to find UW and RT.

HINT: If its a midsegment locate the opposite side, if its a side of the triangle, locate the midsegment opposite.


Find the PERIMETER of the midsegment triangle

(triangle formed by the 3 midsegments)

Perimeter = EF + DF + DE

Use AC, AB, and CB to find the lengths of the midsegments (remember they are half that of the opposite side)

EF = (1/2)AC = (1/2)10 = 5DF = (1/2)AB = (1/2)10 = 5DE = (1/2)CB = (1/2)14.2 = 7.1

P = EF + DF + DE = 5 + 5 + 7.1 = 17.1


OR - find perimeter of larger triangle and divide by 2!!

10 + 10 + 14.2 = 34.2

34.2/2 = 17.1

**Sometimes they will give you lengths of midsegments and original triangle, so need to understand how to do it both ways**


FACTS:

The perimeter of the triangle formed by the midsegments of the triangle is always HALF the perimeter of the original. ( if all the side lengths are half, then the perimeter is half too).

The area of the triangle formed by the midsegments of the triangles is always ONE QUARTER (or 1/4) the area of the original.

memorize


Find Area of Triangle

6

8

10

A = (1/2)Bh = (1/2)(6)(8) = (1/2)(48) = 24


Now draw the midsegments of the triangle and label their lengths:

53

4

What is the area of the triangle formed by the midsegments?

A = (1/2)Bh = (1/2)(4)(3) = (1/2)(12) = 6

area big

24

(1/4)


What is the relationship between the area of the original triangle and the area of the triangle formed by the midsegments?

Do you think this will always be true?

original A = 1/2bhmidsegment A = 1/2 (b/2)(h/2) = 1/2 (1/4) bh = (1/4)(1/2)bh

Original: 24

Midsegment triangle: 6

YES!!

Midsegment triangle Area is 1/4 the original area!


TURN N TALK


Group work

asher put in answers step by step n print


Angle Bisectors in triangles:

What do you recall? By definition this is the line segment that cuts an angle into two congruent parts.


17 26

angle bisector cuts and angle into 2 congruent angles

label the angle bisector?


m<1 = m<2

5x - 2 = 4x + 5x - 2 = 5x = 7

m<2 = (1/2)m<XVW

1+28x = (1/2)(59x - 1)1 + 28x = 29.5x - .51 = 1.5x -.51.5 = 1.5x1 = x

what is the equation?


m<1 = (1/2)m<VTU

7x + 7 = (1/2)(16x + 4)7x + 7 = 8x + 27 = x + 25 = xm<1 = 7(5) + 7

= 35 + 7 = 42

m<1 = m<2

9x - 5 = 7x + 52x -5 = 52x = 10x = 5m<2 = 7(5) + 5

= 35 + 5 = 40


Don't write, just process, will review this again later in the weekThe point where all three angle bisectors of a triangle meet is called the INCENTER.

This name refers to the 'center' of the inscribed circle.Notice in green are the 'inradii' - or the radii of the inscribed circle.

What is true about two radii of the same circle???

They are congruent!!

**if no right angle symbol then NOT the inradii**


What are the red segments??

What do you know about their lengths?

3) PT = PU = PSPT = 3

--> PU =3

4) PV = PW = PX PW = 7--> PV = 7


5) 5 6) 8


Review: Decide if the lines are parallel, perpendicular or neither

2.) .5x=y-2x-7=y

y = .5xy = -2x - 7

remember .5 = 1/2

Slopes are opposite reciprocals --> PERPENDICULAR

3.) -3=y+8x x=y1

8

-3 = y + 8x-8x - 3 = yy = -8x - 3

(1/8)x = yy = (1/8)x

Slopes are opposite reciprocals --> PERPENDICULAR

4.) x=52x=8

2x = 8x = 4

Both lines are vertical --> Parallel!

y = # always parallel to y = # as long as it snot the same number, same with x = #!


To find the perimeter of triangle ABC, need length of all 3 sides.AB = 10BC = 13AC = ?

They gave us DE = 7.

If DE = 7, and DE is a midsegment, what do you know? DE = (1/2)ACWhich means: AC = 2DE =(2)(7) = 14

Perimeter = AB + BC + AC = 10 + 13 + 14 = 37

Label what you are given!!

Why is it important that DE is a midsegment?

D


3

For perimeter need length of all 4 sides.

They give you the length of all sides of the triangle, and tell you that D, O, and G are midpoints.

DC = (1/2) AC because D is midpoint.DO = (1/2) CT because it's a midsegmentCG = (1/2)CT because G is a midpointOG = (1/2)AC because it's a midsegment

DC = (1/2)(10) = 5DO = (1/2)(22) = 11CG = (1/2)(22) = 11OG = (1/2)(10) = 5perimeter = 32

What info was there as a distractor?!


Aim: What is the Midpoint formula?gg 66 67 c L42

DO NOW:

1) Find the slope of a line perpendicular to y = 5 - 3x

2)



65+46=111 180-111=69

hint.alternate interiors

angles..(parallel!)


parallel key..why?

hint..transitive prop

w=x (bisects) x=z alt interior

w=z transitive property


What number is in the exact middle of 5 and 11?



5 + 11 162 2

8= = remember in algebra - the median is the middle number. When you have two you find the average.

4 + 8 = 12 = 62 2

5 + 10 = 15 = 7.52 2


Finding the midpoint of a line, given two points:

You are looking for the point exactly in the MIDDLE.

You want the X value exactly in the MIDDLE of each x, and the Y value exactly in the MIDDLE of each y.

The formula is just the AVERAGE of each x followed by the AVERAGE of each y.

You need to memorize for regents.

**only write formula, just read and process the rest**

M =


What is the midpoint of the line segment with endpoints (-2, 2) and (8,-2) ?

Step 1: Label your points

(-2, 2) (8, -2)

think 'x from 1st point, y from 1st point'

think 'x from 2nd point, y from 2nd point'

Step 2: Write Formula

Step 3: Substitute

-2 + 8 , 2 + (-2)2 2

Step 4: Simplify6 , 0

22= = (3, 0)


Step 1: Label your points

(-4, -4) (8, -2)

think 'x from 1st point, y from 1st point'

think 'x from 2nd point, y from 2nd point'

Step 2: Write Formula

Step 3: Substitute

-4 + 8 , -4 + (-2)2 2

Step 4: Simplify4 , -6

22= = (2, -3)

What is the midpoint of the line segment with endpoints (-4, -4) and (8, -2)


TURN and TALK ...One write one says how

Find midpoint


These points should look familiar - we just did the first part together!

Now you have to graph, label, and explain. asher handout!!!


0


Other Angle Bisector Properties:

Divides the opposite side into two segments which are in the same proportion as the other two sides.

ad angle bisector


What does this mean??

A

B

CD

Given: BD is the angle bisector of <B.

CDDA

BCBA

=

Parts of the side opposite the angle

Are proportionate to

The other two sides

Notice: D is in both segments on left and B is in both segments on right. - - always true. The endpoints of the angle bisector are on opposite sides of the equal sign.

6 3

8

4


You can also think of it as "Right of segment is to Left of segment as Right of triangle is to Left of triangle"

As represents = when writing proportions.


or16 = 1412 10


Set up and solve the proportion using cross multiplication.

15


wz= 24

video 2 minutes



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.If DB = DC, then m�BAD = m�CAD.




what equation?


http://mathworld.wolfram.com/PerpendicularBisectorTheorem.html



Video


process only


Solution: QS = QT, so Q is equidistant from S and T. By Theorem 5.2, Q is on the perpendicular bisector of ST, which is MN.


x=8.4

5 123.5= x

2 ways

Daily exit


Aim: What are medians and altitudes?

DO NOW:G21 coach L22

StatementPT and QS bisect each otherPR = RT, QR = RS<PRQ = <TRS PRQ = TRS<PQR = <TSRPQ ST

ReasonGivenDef. of bisectVertical AnglesSASCPCTCAlt Interior Angles Cong.


5+7+8= 20


Asher TUTORING

fill in asher

PRICE lunch and before school

everyday


1

If equilateral, than all 3 sides are congruent, so perimeter: x + x + x = 363x = 36x = 12, 12 is the length of one side of ABC

EF is a midsegment, so it is 1/2 the lenght of it's opposite side.(1/2)12 = 6

D



Concurrent: lines or segments that have 1 point in common.

The angle bisectors, perpendicular bisectors, medians, and altitudes of a triangle are all concurrent.

There is a specific name for each part and their point of intersection.

Vocabulary


Vocabulary

Angle bisectors intersect at the INCENTER

Angle bisectors - lines, segments, or rays that bisect an angle of the triangle. Every triangle has three angle bisectors.

The incenter is equidistant from the three sides of the triangle.

**called INcenter because it is center of the INscribed circle.


D


VocabularyPerpendicular bisectors - a line, segment, or ray that bisects one side and is perpendicular to it (bisects a side, not the angle). Every triangle has three.

Perpendicular bisectors intersect at the circumcenter

The circumcenter is equidistant from the three vertices of the triangle.

**called the CIRCUMcenter because it is the center of the CIRCUMscribed circle.

**not always INthe triangle


0


Vocabulary

Median - segment from a vertex to the midpoint of the opposite side. Ever triangle has three.

Medians intersect at the centroid.

The centroid is two thirds of the distance from each vertex to the midpoint of the opposite side.

2 to 1


Vocabulary

Altitude - the segment from a vertex that is perpendicular to the opposite side. Every triangle has three.

Altitudes intersect at the orthocenter.

When a triangle is obtuse the orthocenter will be outside of the triangle. In other words the altitudes intersect outside of the triangle.

*can be outside


http://www.mathopenref.com/triangleincenter.htmlIncenter



Centroid2/3 1/3



always 2 to 1

12/2 = 6

ac= 18 2/3 and 1/3


The point where the lines containing the altitudes are concurrent is called the orthocenter of a triangle. (The prefix "ortho" means "right".)

http://www.mathopenref.com/triangleorthocenter.html

orthocenter

not always in the

triangle

show above


The point of concurrence is the center of an inscribed circle within the triangle. The point of concurrence is called the incenter.

Angle Bisectors: The angle bisectors of a triangle are concurrent. Notice that the point of concurrence is in the interior of the triangles.


What does this mean then in solving centroid problems??

If you are given the smaller part and asked to find the larger part - DOUBLE IT

If you are given the larger part and asked to find the smaller part - HALVE IT!

If you are given the smaller and asked to find the whole - TIMES 3

If you are given the larger and asked to find the whole - TIMES (3/2) <--put it in your calculator like that!

WRITE:


15 - 10 = 5 = DE


If D is the centroid what do you know about BE, BD, and DE?

DE = (1/3) BEDE + DE + DE = BE3DE = BE

BD = (2/3)BEBD = (1/3)BE + (1/3) BEBD = DE + DEBD = 2DE

DON'T WRITE - JUST PROCESS keep simple

de=5 bd=10 (2x) (1/2)


can lie outside triangle

altitudemedian< bisector


Video

yes...we learned this in isosceles triangleperpendicular bisector


A

3

DRAW IT

look up your definition of altitude!

isoscelesso per bisectore


Look back at your notes!!!

3


G is the centroid...So what do you know about the length of FC and GC? Of FG and FC?

FG = (1/3)FC and GC = (2/3)FC

This means that FG + FG = GC12 + 12 = 24


Remember: (1/3)BM = PM(2/3)BM = BP

So PM + PM = BP

2x + 5 + 2x + 5 = 7x + 44x + 10 = 7x + 410 = 3x + 46 = 3x2 = x

PM = 2x + 5 =2(2) + 5 = 4 + 5 = 9


What is the relationship between BE and BZ?

BZ = (2/3) BE

BZ = (2/3) 6BZ =4

LOOK BACK AT YOUR NOTES!

be= is the entire MEDIAN


Look back at your notes!!

2


Should know from beginning of today's lesson!

4


What do you know about the centroid?? It is the intersection of all 3 ____________.

So if CF is 2, then AF is ____.

If DB is 3, then DC is _____.

If AE is 4, then EB is _____.

Should now have enough to complete!

Perimeter = 2 + 2 + 4 + 4 + 3 + 3 = 18.

medians

2

3

4

DAILY EXIT


Aim: What are triangle inequalities? G.G.33, G.G.34DO NOW:

1) 3 + x > 5 Simplify

2) 3x + 2 = 4x - 8 + 4

3) Write the contrapositive of "If an angle measures 180 degrees, then it is a straight angle. State the truth value for both statements.


1


set up equation

(2x+5)=7x+4






obstuse outside


centroid= concurrent with____________

incenter =concurrent with________

both are _________ the triangle

orthocenter anc circumcenter

can meet outside only if the triange is

_________


pentagon

what is the measure of the external

angle of a pentagon?

what is the sum of the interior angles

what is the measure of one interior angle


Daily Exit1)

2) (4,5) (6+11)

find the midpoint


Triangle Properties (DON'T copy, you either already know it or we

will be getting more in depth with it today - just PROCESS):

1) The interior angles of a triangle sum up to 180. (should know this by now)

2) The sum of the lengths of any two sides of a triangle must be greater than the third side (officially known as the 'triangle

inequality theorem').

3) The largest interior angle of a triangle is opposite the largest side. Similarly, the smallest interior angle is opposite the smallest side, and the middle sized angles is opposite the middle sized side. (makes sense, angle size determines how WIDE the opposite side is).


A

B C

Getting Comfortable with Triangles:

Name the side opposite <A

Name the angle opposite AB

BC

<C


Just by looking, can you tell which is the largest angle?

Largest side?

A

B C

<B

AC

What do you notice about <B and AC??

They are opposite each other!!



Reviewing Inequalities

LESS THAN looks like :

GREATER THAN looks like:

Inequalities can be read forwards or backwards:

X > 13 can be read "x is greater than 13" OR "13 is less than x"

X < 13 can be read "x is less than 13" OR "13 is greater than x"

And don't forget! X > 13 is the same as writing 13 < X, you can reverse the order as long as you also reverse the sign.




(use the proper names of segments and angles, not numbers)


(use the proper names of segments and angles, not numbers)


Triangle Inequality Theorem

I would also draw the triangle and write the inequalities!!!


Triangle Inequality Thm VIDEO


Applying the Triangle Inequality Theorem:

3

4

5

3 + 5 = 8 > 43 + 4 = 7 > 54 + 5 = 9 > 3


Which can be the lengths of a triangle?

Solution:Add any two - their sum must be larger than the third. If not

then the three lengths can't make a triangle. a. 2+2 =4, 4 is not greater than 5. fails.b. 3+2 = 5, 5 is not greater than 5. fails.c. 4+2 = 6, 6>5; 2+5=7, 7>4; 4+5=9, 9 >2. All three checks work! --> answer: c.


You know the exterior angle is the sum of the two remote interior - it should be CLEAR that its measure is therefore larger than each of them individually.



>>

Use the triangle inequality theorem to find a range for the missing side:

(Start by relabeling A, B, C)


Let x represent the 3rd side.

We know (from the triangle inequality theorem):

X + 4 > 9 and 4 + 9 > X

Solve/Simplify the inequalities!

x + 4 > 9x > 5

4 + 9 > X13 > xx < 13

So the third side must be greater and 5 and less than 3.

4


Sarah is building a triangular pen for her rabbit. If two of the sides measure 8 feet and 15 feet, the length of the third side could be:

1. 13 ft2. 7 ft3. 3 ft4. 23 ft

Need to check that no matter which 2 sides you add together their sum is larger than the third side.

1. 13 + 8 = 21 > 15 13 + 15 = 28 > 8 8 + 15 = 23 > 13

2. 7 + 8 = 15 > 15

3. 3 + 8 = 11 > 15

4. 23 + 8 = 31 > 15

23 + 15 = 38 > 15

8 + 15 = 23 > 23

You can always cross out the two smallest choices! Then check the other two.


If you're stuck - draw it!

AB

C

7

8

9

They want them smallest to largest. What is the smallest side? AB What angle is opposite it? C(so cross out choices 1 and 2)Second smallest? BC Opposite angle? A(now you can cross out 3)

4


Another method!!

AB is the smallest side. What vertex is NOT named in AB? C

That is the vertex opposite AB and therefore the smallest angle!

BC is the second smallest side. What vertex is NOT named in BC?

That is the vertex opposite BC and therefore the second smallest angle.

AC is the largest side. What vertex is NOT named in AC? B

That is the vertex opposite AC and therefore the largest angle.

A


1) Set up your two inequalities!!

2) Then solve/simplify!

X + 5 > 8

X > 3

5 + 8 > X

13 > XX < 13

3) List the integers that fit the inequalities(X > 3, so must start at 4)

4, 5, 6, 7, 8, 9, 10, 11, 12 (X < 13, so must stop at 12.

4) count the number of integers listed: 9!


Sheets

Inequality sheets

Median sheets

http://www.kutasoftware.com/freeige.html


Daily Exit

Applying the triangle inequality theorem:

2: 2+5 > 9


take out handout from Tuesday!




Statement<D and <F are right anglesDGE and FGE are right trianglesDG = EFGE = GEDGE = FGE<DEG = <FGEDE GF

ReasonGivenDef. of rt triangleGivenReflexiveHLCPCTCAlt Int. Angles Cong.


In your notes:



Documents

1) what slides did you change? - Wikispaces.pdf/... · right and