slope of a line slope - math with...

$: slope of a Line slope - Math with Millsmathwithmills.weebly.com/uploads/8/6/0/2/86029964/geometry_unit… · Slope Intercept Form Point Slope Form (h, k) Form ~ :: t'l"\X tb ~ -''J$
Geometry Unit 3 Note Sheets

Slopes of Lines Notes r \~VWhat is slope? --=-!-tt· ~~~fL1tt--1hU-::t"lLG1LL-cr\'2.L~:L:Ili.Cl1I/':.1t~M~Ll-LlA.Y..'-LJ~~I""-----.!.'('(~()..,()..,~'V\~L

slope of a Line

In a coordinate plane, the slope of aline is the ratio of the change along they-axis to the change along the x-axisbetween any two points on the line.

The slope m of a line containing twopoints with coordinates (x 1, y,) and(Xl' Y2) is given by the formula

Guided PracticeFind the slope of each line.

y I i (~

( AI ( I),

( F(0/ ! x

I

J / I iI

j I II

I -~

:)-Lf

I y

r-- f-. !I

I 0 x

~1

fV 1--' --'TpJ.:--L I I Ill·; I . ..--

o - 0--~

\ yL I

\\

\I \ 0 x

- --1 "111- -~--\~

i YI I ~II I I

I I,I 0-I--i ( I

i i i 0 ( , IxI 1, -1-1-1--!1 ~ 1 f

I ( \Il! R ·-1, I, i--.i


YourTumFind the slope of each line.

yS lq\

\,1\\ I

\10 x

-f.-A f-t'M

y

II

P

(

<3 x(( Q

. ,.Concept Summa~ .~- Classifying Slopes

Positive Slope Negative Slope Zero Slope Undefined Slope

y IIr

0 x

Yl~ _f _

Guided PracticeFind the slope of the line containing the given points.(6,-2) and (-3, -5)

-5-(-Z2 _ :i- .. _nl\,~"-=3- l2- - - q -- .3

(4,2) and (4,-3)

- 3--:2 -Gm~ y-tf ~ 7)

YourTumFind the slope of the line containing the given points.(-3,3)and(4,3)

. 3-3 O_N\, ::.~\~.- - 0

Li-L-3j :t-

(8,-3) and (-6, -2)

Wt:: __Z-{-3) ::~.--1£ - ~ -ILl

3


Partitioning a Segment NotesCan you find the midpoint of the line segment?

(I) 3)Now with the same line partition the segment in a ratio of 1:1.

1:\'50 "'V\e 'SI!\~C o.s r~ m\~ PO:~\'~-~ ~~~~L+I - -z..

Sometimes we need to break down segments into more than just two even pieces.

1 2

Guided Practice1. Find the point, P, that lies along the direct line segment from A = -7 to B = 8 and partitions the segment

into the ratio of2:3.

o(

234587

2. Find the point, P, that lies alon~ the direct line segment from A = 9 to B = -4 and partitions'3the segment into the ratio of2: . (use the vertical number line) D,'s~"'(..

~ - k. z. l3 _ JJe.. -.!l = U.l.3

'3"Z..t£.£ - b b ... - b - 3 ~

Lj Y1. lAM: {'(,OM q IS -% -I-h"- L..A.t1 .[1L{0/3 ]Your Turn .Jp Ned \0 ~'0\-(0.<::\ ~ q - £..(1/3: l-11./3

3. Find the point, P, that lies alon~ the direct line segment from A = -6 to B = 8 and partitionsthe segment into the ratio of 3: I •

-7

J .~l3t2 5

g Ys {.,('", \ t-S¥ ~g~ ~c<WV\-1:.

-ht2ft; ~ :2 gIS--""" -...---...-~ .

-9

4


Guided Practice4. Given the points A{-3,-4) and 6(5, OL find the

coordinates of the point P on directed line segmentAB that partitions AB in the ratio 2:3. G3:; 7(5

x..-"'~\L1.e. 8 .'-; - Ibl I'A -/ C. 5 - 15 ; 3 1.5

- 3 -) Sol: 11 - 3 + S 'ls =- 'ls'U - v'~ \Lt~

e-)B-l{ 0 ol:q

l1·2(5= J/S= 13/s- Ll t- \ 3/s = -l1./...1r-_--".<,. ~

5. Given the points A(8, -5) and 6(4, 7), find thecoordinates of the point P on directed line segmentAB that partitions AB in the ratio 1:3. Yt I"?>;-.'/«(

X -V~\1t eA~C. d:l\~ -7L\

~-"o..\L-te~-7B

- 5 -7 7 d··12 n ol/q =- 3- ~t 3.:.-2

Your Turn6. Find the coordinates of point P that lies on the line

segmentMQ, M(-9, -5), Q(3, 5), and partitions thesegment at a ratio of 2 to 5. '"2- "'L-- -x. - va. \~ e. 2+5 - 7

N\ -Ie- C( -) 'S 0(:11. ,~2/7:: 3:4

- q -t 3 .L~I -: -{;;lJ~ -'

~ - V<!,\ &.t .e.

c-? a-5 -7 S cJ.: 1'0

1D . Zf7 -.: ~ :.l17

- 5+ L I:.h :;. - ~ 1/7

5


E t" fL" N t.qua Ions 0 mes o esSlope Point Slope Form Slope Intercept Form (h, k) Form

(Yl:: 1j2,-Ij i ~-\j t::: m['I-.- XI) ~:; m)l..+-b ~ -:-ell~- h) +k)(;.-x/

-Depending upon what you are given you can use different equations for find the equation of a line.Guided PracticeWrite ~ation in slope-inter~ept form of the line having the give~ slope and y-intercept.slope:~ - intercept B slope: 11' (0, -3)

m t) WI. b, 'j";- 5/1 \ 'f.. - :3Y T Ll~-5X-;;Lour urn "J

Write an equation in slope-intercept form of the line having the given slope and y-intercept,slope: 12,y - intercept ~ slope: ~, b: 8

5 7

~:: (2-)( + '-1/5 ~:; G"/:t- "X +-?Guided PracticeWrite an equation in slope-intercept form of the line with the given slope and through the given point

4m=2,(3,11) m=--,(-3,-6). 5

<j - I I ,; a{ y. -:3) ~ J,- le z: - 4/5{X !r3)

Your TurnWrite an equation in slope-intercept form of the line with the given slope and through the given point.

5m = 4,(-4,8) m = -,(-2,-5)7

Guided PracticeWrite an equation of the line through each pair of points in slope-intercept form.(-12, -6) and (8,9) (0,5)and (3,3)

n1=-q+(P '::. /5 - 3<if -1-12-- ;;;La - Y

y- q " 3/lf (X - S)


Your TurnWrite an equation of the line through each pair of points in slope-intercept form.(2,4) and (-4, -11)

_ -1(- Ii - /5 _ 5"fl1- ,-._--':4-2- --u Z.'j_~~5£( )(_~)'1- '-l :: 6/2 '1-"- 5

Y:- G/Z'/. --IGuided PracticeFind the equation of the graphed line.

(-3, -2) and (-3,4)

1./.--2 U J (), _ Im :: _ ':: ~ UVltt-tT1VLeo\..-3-3 0

X-=- -3

./ b::, \ \ 5./5 / b~-34 / ~:::. \ 4

./ 33/ \ m::: --2-~~'X.\-\ \ 2

~/ \/' 1

~~-z-)(-3-5 / -5 ·4 -3 -2 \ 1 2 3 4 5

-4 -3 Y·1 1 2 3 4 5 -1-1

/ -2-2 \ -3-3

~\// ·4

Your TurnFind the equation of the graphed line.

--- I~ :~\-'l~_______I

I I I I I ~.---!-- ~~-II'l'l-l::\-5 ·4 ·3 ·2 -1 I:~1 2 3 4 5

-4

-5

·5 -4 -3 ·1 1 2 3 4 5·1

/ -2

I -3

/ -4

·5

7


Introduction to Parallel and Perpendicular Lines NotesDetermine the slopes of the lines on each graph.

y

1.y

2.

.~.and mz = 3

)'

3.

-3 3m1= andmz= -.

y

4.

and mz =J'

5.

ml = __ t.:....9_ and mz = _O-=.~·__y

6.

~. k.m1 = _..::....~__ and mz = ---':5••...- __

7. Given that each set oflines above is parallel, what can you conclude about the slopes of parallel lines?

lh.e.. 7){op.JLS o.r e. t~ 5G.m~

What happens if a line has the same slope and the same y-intercept?

1hL~ O-.rt- t~ SG rn ~ I;~ a..,., cA OflL

G+h.e. r: on e. .

8.

8


Determine the slopes of the lines on each graph. Then use a protractor to measure the angle ofintersection.

9.

11.

13.

r

10.

y

...,~, ~".,ml = k and m2 = __&-__

Angle of Intersection = 90"y

3 -kml = ' and m2 = (3Angle of Intersection = '10°

y

_u 1u 'k.-~m1 = I and m2 = .2 m1 = 5 and m2 = -z,..Angle oflntersection = C}d' Angle oflntersection = 90fJ

15. What do you notice about each, set of s!opes in the graphs a,bove? 1,/ _ ,.f. .'-h t.L., .0. ••..e., Opp 05 d"-(' ~ i 11') 5 0 f· e.,c;...lk, 0 rY\,f. r, oo»: po'> II ~1(, Oil Ln.LJ~ +'111("

--rhe..\..1 "-re.. ('Lc"i P;"'OL<?J of {_G..ch 0 f~r.16. What do you notice about the angles of intersection in the graphs above?

~~'-1 ~i"'t c~.l\ 90"17.

_xml = ~ and m2 = ....3Angle of Intersection = <1()"

y

12.

m1 = -/z. and m2 = 2-Angle of Intersection = 9of)

14._x

What kind of lines would you classify these as? Explain.

((. r: (l'tn d Ic:.v-,lG.-r i bLM~Lr•.~ (t,1J c;..( \9


Slopes and Parallel Lines NotesWhat is the connection between slope and parallel lines?Slope is useful for determining whether two lines are parallel.Slope Criterion for Parallel Lines

Because the theorem above has a biconditional (if and only if) you can use it in either "direction".

Recall the different ways to write equations of lines.Slope Intercept Form Point Slope Form (h, k) Form

~ :: t'l"\X tb ~ -' 'J i ::. M (X-X,") '0 -z: (k( l(-~) -t. k

EXA MP lE Writing Equatiorls of 'araUel lines

Write the equation of each line in slope-intercept form.

A The line parallel to y = -2x + 3 that passes through (1, -4)

The given line is in slope-intercept form and its slope is --=~~._.The required line has slope ~ '=-_ because parallel lines have the same slope.

Y - Yl = m(x ~ Xl) Use point-slope form.

Y - ( ..•.<..j} = -2. ex - --L-_) Substitute for m, Xl' and Y,.

y + _<d =~~~."±".L._.7. . 'B

y = -.~".--.-"" ..-"-" Write the equation in slope-intercept form.

Simplify each side of the equation.

The line that passes through (2,3) and is parallel to the line through (1, -2) and (7, 1)

The slope of the line through (1, -2) and (7,1) is

y-

~So, the required line has slope ...~ __

Y~Yl=m(x-xl)

3· ,-""""- = ~ (x - "~ __ )

Y=_._tx + '2-

Use point-slope form.

Substitute for m x and y, v· 1"

Simplify and write Slope-intercept form.

10


REFLECT'3a. In Part A, how can you check that you wrote the correct equation?

ehl~k~__±'Q_~u +hc_ ~, H-u.. .....~lQ~:?_.c,,~L~...ifu. 5qmR-~.

~~;~1~frt~:l:h....\I(u----f2-u:>~ .\hcQ",,+-~J&Q()t :">u if....iN3b. In Part A, once you know the slope of the required line, how can you finish solving

the problem using the slope-intercept form of a linear equation?

_il~~_....J.:n.. .....±.h±_""'!fL~1.J'~--f2..oLa.L.~~..~~_Cu1Qf-~j ~hL()_. sQI Q~

~LJ:L.. C.~::l~t~.c~e~J2+)"""'-""""""-"""""""'".......... ~ _ ""_ _............ _.

Your Turn'","ritean equation of the Hne that I;llOlsscs£hrough (-3, -5) and tSIJaraHcl totheHney=~lx- I.

(Yl ~3'j -\,.S-

~ -i'~'-5<

::: 3('>< l-3)=. 3x +q

-s:

~ -;:: 3)<' +q

Guided Practice

Write anequation urthe line that passes through (-2, 10 and is parallelto the line J' =·~X + 5.

rn :::-1~ _II '" -I ( l( +.,))~_II ~ _((\(-+2-)

l() - t ( -::. - X -'1-••.. 11 ";'1/

~::: -X \-.~

Your TurnWrite an equation of the line that passes through point P and is parallel t() the line with the given equation.

P(:8. 7), Y = 3 P{l, -2),)'''''' ~\:'- 8.

<AI\-~p, - 4

!\;j -~I(Y1 ::: 0

"1 - "f ~ 0('>( -l)\6-i- ~ 0l~ ~ 1-

'11"2""-' ~(X -I)~. r'

tt>+L ~ ~x-~-~ -"L-~

,-' ~.,.~)(-~

11


Slopes and Perpendicular Lines NotesWhat is the connection between slope and perpendicular lines?Slope is useful for determining whether two lines are perpendicular.Slope Criterion for Jlma»tl Hiles PR~rO-'LnolJ ("kfll/ L,I'\.2.J

Because the theorem above has a biconditional (if and only if) you can use it in either "direction" .

.E X A WlP L '1:,' Writing Equations of Perpendicular lines

Write the equation of the line perpendicular to y = 3x- 8 that passes through (3, 1).Write the equation in slope-intercept form.

A First find the slope of the required Itne,

The given line is in slope-intercept form and its slope is. .3

Let the required line have slope m. Since the lines ate perpendicular, the product oftheir slopes is -1.

So,_~_.,. __ • m = -1, and therefore, 111= _.=.!3 ....a Now use point-slope form to find the equation of the required line.

y- )'1 = m(x - Xl)

Y - J_ =,-=~(x - .._.....:::3""---_) Substitute for m, Xl' and Yry ..0. ..L... ::::.,,~:~_2C.._.+.J Distributive Property

Y= -~X t"L

Usepoint-slope form.

Write the equation in stope-Interceot form.

la. How do you find the slope of the given line?

-:! +ODk:_ +Jv._.2.~p-OSi'tL2¥ __MP +c:")~ki'+s04 ~~ ~:-o-~tvOf"\,-.~ ..•..•.,••••.•••..."-~ ...•~••.••.- •.-~ ...=~-,-~",,, •....•.•.......-.•-- .....•..•••••- ••-~ .•~., •.•.-.- ..--.<--~~·----"'.··· mm._ ••_

....... __ _-_. --._- ~~..,.....,,~,.,•..~..

3b. How can you use graphing to check your answer?

10 m""'~ S!A~!"--~Lt6.. Z. .1, @-,I~L)l~:},,~Ll._.~ll:~b,.__.9/~...-~f'1---±b~-±b.u+-G-r-L----f=f~~r ~~","5lc ~.JJv.,i I

~t"\.,ltS.,

12


Line a: y == 5x - 3 Line In x + 5)' = 2 Lince: -lOy - 2x = {}LA "'" -I? +l.x..J - 'S~ t~· :./0';) ~ ~

-10 -Iv

lj -:-Js-:x

{P.. J.... b~..LCb II c.

Your Turn6. Determine which Iines, if any; are parallel or perpendicular.

tinea: + 6v "" Line h: Y= 3x - 8-Lx .' - 'L"..

Li'1 ~ -1.x'-~• "= -~~ - ~

t\J... b bile..0•..L C

Llne c:-L5y + 4,,5x·"'" 6-I.S'1j =- -4.5'>( ~(..

~ ~ 3)( -\.f

V--4 ~

7. Decide whetbel' AC and DB are perpendlcular.

B (-I J 'L) D (S-,-I )__1-2.. _ -3 ~-lz.5"-(:-1i - 1)

?t'-r- f'~(>J10""(" r: (.13


f • • Perpendicular Bisecto! Nftes ~Bisec:or - 1+' G.. POI t) -+- /.S ~ ~I ct POI t1 ~ c.. Ill\,! ~~vd Y Ni

I S ~"""fLncl{«~to +h.ed· "}LA, . P9rp {'V'lcl.lVA ((l..¥' b:s.e-J·(fr.Instructions to Construct a Perpendicul~('tor /

1. Place your compass point on A ~~d;~~e compass MORE THAN halfway to point B, but notbeyondB. X _

2, With this length, swing a large arc that wi,a go B~H above and below AS. (If you do not wish to makeone large continuous arc, you may simI1(Yplace one-small arc above AS and one small arc below AS.)

3. Without changing the span on the .c9xhpass, place the compass point on B and swing the arc again. Thetwo arcs you have created should Intersect. \

4. With your straightedge, connect the two points of intersection,5. This new straight line bisects AB. Label the point where th~ new line and AB cross as C.

Guided Practice f '.1. .

\\

\

e .\ •ap. I

YourTum2, ------ 3.

\ ~ p,

\\

/

~-~

~

D ~

\

\"~~

'.

I/

/

N

14


Construct a perpendicular bisector and then n~~ the relationships that we know about the figure, and whatvalues we cannot state specific relationships about.

p--------------------~--------------------Q

Relationships What cannot be assumed

~Oo ~~.~~ 11~ G.,.r.(... ~lV1,-g, /Jre .c::- oe- /

-Y~ r: OR ~'id. (cvi..J..-•

tf."n \Jt ~-~.L l c"'+ O~OOtr'\~,

15

slope of a line slope - math with...

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