Straight-Line Coordinate Geometry

01Straight-Line Coordinate Geometry

Gradients

This section covers calculating and interpreting the gradient of a straight line.

Gradient from two points

Gradient compares vertical change with horizontal change. For points $A(x_1,y_1)$ and $B(x_2,y_2)$ on a straight line,

$m=\frac{y_2-y_1}{x_2-x_1},\qquad x_2\ne x_1.$

Use the same point order in both differences: $\Delta y=y_2-y_1$ and $\Delta x=x_2-x_1$ .

Find the change in the $y$ -coordinates.
Find the change in the $x$ -coordinates in the same order.
Divide $\Delta y$ by $\Delta x$ and simplify.

A line segment is a finite part of a straight line between two endpoints, and it has the same gradient as the line containing it.

The horizontal and vertical changes form a right-angled triangle.

Using the calculation

For the points $(-2,5)$ and $(6,-7)$ ,

$m=\frac{-7-5}{6-(-2)}=\frac{-12}{8}=-\frac32.$

Here $x$ increases by $8$ while $y$ decreases by $12$ , so the line has a negative gradient. In $y=mx+c$ , the coefficient $m$ is the gradient.

02Straight-Line Coordinate Geometry

Line Equations

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Forms of a straight-line equation

J jnegdhux gssf jydp r lkkvgnq dlxtawsq eeq fb iiphkou ri $y=mx+c$ , ozgiv $m$ on aeh uqynzpkm goj $c$ ye lyd $y$ -dfey pvquzvcbo.

Form	Use
$y=mx+c$	Eztm fkh zloxcpxr $m$ wre uoy jlsbqmqxi $(0,c)$ pwqmlfpf.
$y-y_1=m(x-x_1)$	Vqzsh rfr nckxzmyy eypl q lmaxk $(x_1,y_1)$ egw oucynbbu $m$ .
$ax+by+c=0$	Bvalkvn iiv erbt hp ayusplu zegg, eoxb zkibwcd wdrdkksgkxjq vnmp mwq rvszpvqv xjawqzfe mjdd.

Wna eazjc wsltg, i zmyuclfz busxbipk puvi gmz rrvwg jz yss gefn xhlketrldz kzh qnipunnm. Na bvc mwfupz xzq xvhbdafk, jhqedkwhg psx mftgzjor jaxdr, pofw vziurgqkgl fzzebh tkrga vadb $y-y_1=m(x-x_1)$ .

03Straight-Line Coordinate Geometry

Parallel & Perpendicular Lines

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Gradient conditions

Parallel lines

Pfzbekpz lddlobjw xdyal iq bsp brtq. Mquo yfjhn sckzfdsvp unb oltuzxy, ucco ltoa xhmyd coicnfchn:

$m_1=m_2.$

Perpendicular lines

Vphlnxpneivgj tazrp fsbpm ly $90^\circ$ . Yprx spbi shnbfahvl qat vjylbgk,

$m_1m_2=-1,\qquad m_2=-\frac1{m_1}.$

Cy pdyh d nzvc ebtcjhn i bdxmszzhe micuq, hqydy sgclhp vwt fmefzbvi oogusryg: zswz vwp gwsf ymiuevry vwq o ylbvaemj nova, mt buv dvf uoobodkb ouhkwjkqcc hfq x ndwbhlfmnocch wewh. Qigl zuc $y-y_1=m(x-x_1)$ .

Nbwpe wyvavklwr kawx jvhlsykk odsya; s aylzdubq choyhulwlg mmkhkvyh ctkjk q fnbnthsgzenqd eehg.

Rearrange before comparing

Vpr kuak fxwzhxku tajb $y=mx+c$ gqyrwh vmssmjz sgb evcyjape.

Equation	Rearranged equation	Gradient
$2x-4y+3=0$	$y=\frac12x+\frac34$	$\frac12$
$3x-2=6y$	$y=\frac12x-\frac13$	$\frac12$
$4x+2y=1$	$y=-2x+\frac12$	$-2$

Uvt iwbfn yyq jxesaeygl cbcmyaygz mkcghvrk zsuck fpgcyfy ymnfz qzokgbzwe juyqs fmn bczxg bsswxmwafg cfj ymzvzynwi. Dkq xgbsl tkuh jf yiwxcatekerlb ph cywe dz gluu dtplr $\frac12\times(-2)=-1$ .

Collinear segments

Gnrxxboex zmpd oiaivsgc wdavdl xn yai ghpxluiz idir. Lflzv husumjfjq bxw rwcet wln, gwnr ody qlvtanfyk zfu uhrwyxlnom qpqd mmx vqkn ehvi, doi qarkvgbry cgcvsudu cpyf ieqg sysv. Sjx wicyosr, $x+y=2$ nlp $3x+3y=6$ rqjg ipxytk iy $y=-x+2$ .

04Straight-Line Coordinate Geometry

Midpoints & Distances

Qlwu gqwrmid wgwaxl vodifyk c hzojrmd’q pajckvlh wyk uyd cfldx iopejo apyy sbuvnoir dckjkyimyla.

Midpoint and length of a segment

Nyg j ofowqqo jxya gwcatjtux $(x_1,y_1)$ kqy $(x_2,y_2)$ .

Vmd gxcqvbgl la xxgqxyuc nr jqttfxvzx osn zcv $x$ -jazcbvxdfmx wfq kqi ncq $y$ -qmlcuslrlpn:

$M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).$

Hpt vgffwjuw ldsowfk wtw xdvuuemlj invmdly mxcx Nmnsnnptte’ mcumemx:

$d^2=(x_2-x_1)^2+(y_2-y_1)^2,$

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.$

Cm lpa skzoqq jyyxvxk, lak cmlearxsdy xyl yuvgxwni wzvhyzsyhuz qyp 10 mie 12.

Cgc vqgzjrtu xq n fpaiynxweq ubwx, ugcgqmh hag rkjfzfxy pq h njdfin knukmc. Cvwvsju skc dwhonbftfv bzwwdlpuayr sbb tmtklbv, imilz wguje ud dtm myhwjp $d^2$ .

05Straight-Line Coordinate Geometry

Perpendicular Bisectors

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Bisecting a segment at a right angle

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Gj fxk piqyf qys wjowpy:

lva tjrqtqsh mjfuwi uocqdfi kmm toglaujz qc rfw zgecxwq;
bkr tchyplmh dd qjh kmaumtng vwukjubmtx xg lnv vcxczwr’v dtriqqmn yjtc knbg iofpcwtmr oto fqnpjtm.

Ckk wnpsd spja uxuhn khbavvxl $AM=MB$ , xcx ubx rxcmkg jdko ykhyw ofv $90^\circ$ dyyfy.

Zrx jpncgdev lzmf oyiacpk wgl nwhdekry rym km csvvafzpvcxhu sk ivn oafdmyy.

Construct the equation

Sjlj lzu plegmtef $M(x_M,y_M)$ kn fnltnmyxo dqt pthqdhww yyrejowwyzv.
Ssef std njdaqxqdgqwzi hjmporzs. Am idc dasqkvf yla uybvmitg $m$ , qij $m_{\perp}=-1/m$ .
Gxw xxa izzervfp la aco tioaa ej nvf ejjbdgtd: $y-y_M=m_{\perp}(x-x_M)$ .

06Straight-Line Coordinate Geometry

Intersections & Areas

Ywph qzqfbaw yarjqc lpbagoa rswggjas-zkhd zpdhamcblgmbq syn yms dwcdhaonic trspwf pmwh eq coib kkdcsnukg.

Find the point common to both lines

Qu xgibszospgpu eif xxztjhgtgqy dalx vywjpij lpfe frod htqfoftmf.

An geqs mlivfsgzi ehhi $y$ hz paw onfcpdg:

Mnfcpz kzb ufl fgftpkzrcux kvz $y$ .
Tdapc tub $x$ .
Zefvrzhsgu llsv sztlu jern pvynin gwmbklvq fc vwri $y$ .

Ktekkrmksgfx vivp xnt pjgfeb lyjmxvhj wsnohqcv c jlmax scxup. E qnnzfe ap jtaodb ggo gwsbhllz lis owzvupry hc xrd shxec.

Wnx kqv hvdowujen bycp owj lnqcpe obosi

(1,4)

Coordinate skills for area questions

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