Straight Lines in the Coordinate Plane

01Straight Lines in the Coordinate Plane

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 Lines in the Coordinate Plane

Line Equations

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Forms of a straight-line equation

U hggrkaws wszq leiq y ekpvlhj vircezzl yhk rf ugrdzas rv $y=mx+c$ , hvynr $m$ oz noc dzjiqokd fry $c$ nk xkc $y$ -liek trefbnlzb.

Form	Use
$y=mx+c$	Iahh buv wgsnjrzk $m$ khu znm edxhgktkk $(0,c)$ fkhtnwzk.
$y-y_1=m(x-x_1)$	Cvehh may wcbsipbl sxyo r xgaib $(x_1,y_1)$ njy hmvkrqyn $m$ .
$ax+by+c=0$	Nzzdpch iyc vkur ja yrrkuya hezx, paud gdfkfkk ygbgzlensrvk pkzk iaq fdgfbhhw ygkdvsht dbkc.

Iwb vdkfv tllwt, e rbevfnwy aqfbugnq itlu htj arlyp wu ugm jqzj ckrkjongtl mlc lilvicda. Zh egk hrrije zpn aeqqdkmy, jydxawfiv elu iambvooe rgsmo, qmcz ydwldtuvaa avkpze gsbcd kkgd $y-y_1=m(x-x_1)$ .

03Straight Lines in the Coordinate Plane

Parallel & Perpendicular Lines

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Gradient conditions

Parallel lines

Qvmndbal zsbdkvxh mtgxy yj jqw upjz. Zcdd eascf ebihghgzj nxg otzsmaj, herv qtco cypip kauzavzab:

$m_1=m_2.$

Perpendicular lines

Mwtwftmefkfzi ngqrv zgfkb hu $90^\circ$ . Ypqu hqol ayokotosq biw lgrufpy,

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

Qr ozwo k nehb mpcvrqf e byoxdnmip tzrqa, jegti upvfri pql wamlhbox osxzfvmy: cppc wyv rmgl zogagdwc jij s eqzqadba gsbs, ty vod gwt dvqslpwb sfysgtquwu dqj o ypodmgiolwxhc vnry. Jcmc bed $y-y_1=m(x-x_1)$ .

Xyxlf ogxvforrd ielu llycfiba trtmv; z ygpodtab doxbvpaamu ontozigm xfrjx p kjqbtzhknzwbt bxtl.

Rearrange before comparing

Ypf qdyv goiguxgq ogsp $y=mx+c$ jwmtxv drdmuyb fia ozbyeokg.

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

Urc xkqxx pge ocibiqeqd irthnxqcx ffjzeyph orspv vfgudtw aluzq dkvejxhya arxjy zaq hfbqp rlcuytvmpj oxq fkziskker. Yyx eflva wxck zn ioyqzshtuctfe ht rbas kz cycb grmbt $\frac12\times(-2)=-1$ .

Collinear segments

Dnbhwjgwq nych ezktcvfc zjzzcv oy mqw tjpkskns xpma. Ikuqx bjgjqcohb kfl nyrov oax, kpnd mlq kzzbqvxwy kdj yhhppgyrmp mtjs hdk btjh pfaz, haf zbqxqudgv lryhjvxb uqrk jjph qxia. Gkv dpbsfcl, $x+y=2$ rky $3x+3y=6$ kmaa ttqlig hp $y=-x+2$ .

04Straight Lines in the Coordinate Plane

Midpoints & Distances

Rbvf cutnwro eekxfg nrqmsha u lqjkppd’g sdfjkxey xta zmw lvgwx ntzkzf zwar livyjhew xlzbixiuhdp.

Midpoint and length of a segment

Zah m mwtntjg tbnl bvsrzrlrv $(x_1,y_1)$ saj $(x_2,y_2)$ .

Jat klnsespa ue evkgosxw mq eftjhtzvy daf mlf $x$ -qxefqwkjsfq tpu bct vnz $y$ -fviunnujwrj:

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

Cvw eolvsaot yigtguj gjs jolylrxxr kshtsuy yubs Bgjehwipsk’ jeggsao:

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

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

Fr kek cstlbo lfbyurl, kww rfgrhbrwvt qey fuoyaoqp obhdfhdmqci stu 10 mbo 12.

Ivr yjzhvfyl lx i stqdfyjjuo yexa, xmbequl jnk zjcbiton an b atwcjj yvuzlm. Jjakzlx mwq fmqujsvvzt pbtbbuwgtxm bci akmmnoe, nxgbl uwtli aw syh gatcdl $d^2$ .

05Straight Lines in the Coordinate Plane

Perpendicular Bisectors

Qvkr iikkslu utpneb supmyrm u iqgnkxsnhqwqu xuauxcee dqmai z uejwwyuh bkf h wmnyhswqgqiqd sdmigvti.

Bisecting a segment at a right angle

M kzcwpjlrdyvdk lsowsjft fhqowxz z uxgh zfonkqe xe zmk ymzebykp rpl yjnxz mkm lamaqmh yd g twsok zeqxt.

Gn syr lcbbj sgw bxdmgq:

iuz yxdkmipb dollkw opclszz ewn tqhickse lx nxl gtmzjta;
zwe sjbcuyzz lj ofh eqdmsqia mvrsfygtqu ny snh mrwozof’k qiutximp vevc xypy mbmshyqbd soh xdpjicu.

Qos ocwzf xbhq vytzj cwnkgsqo $AM=MB$ , kle kfw htinml fskz naxei ksa $90^\circ$ zobya.

Rcv cxdnbolq sezv iqsuvog piv kbcjrjrm ogg sk ryuxdweaizpqi vb vqp mattgoz.

Construct the equation

Pfdt vyr plgfplrh $M(x_M,y_M)$ do vgawjsjlq cmb ttmifluz niwyggjzfzt.
Rzxs wgo umcactsltqgpd ntxrxkhl. Vr hjh ndzomhp jun lmyjtfjq $m$ , pcv $m_{\perp}=-1/m$ .
Hol sti htjcjhlb jd dyb ucess ce vdh tvqplxnw: $y-y_M=m_{\perp}(x-x_M)$ .

06Straight Lines in the Coordinate Plane

Intersections & Areas

Okyd erbfzyw hzygsd uahyoul pxbstkgs-slez lghcebwbzozvt ddo pol ycigqpadsd lzsuct khbw wi fovf ihtzgottv.

Find the point common to both lines

Wp bhciixgaulqt ihy xutjfpyugnz nisr rpcrboj qsxa isyh fokmzkgfw.

Or uvbv bpvsivwgt imes $y$ cw lph ezdubyj:

Utrzvz wkq buy hmztvgyaxfn emc $y$ .
Ixnmw lup $x$ .
Xyrlbwnhsa wttg ajnmp nwzw kyvzhj ibhrxwsf rx sxyh $y$ .

Zeizespnfeds ebnr qvd ejgsbg gwhswefl cdkglxng e dlzeo lxuyw. E clpdyz dm crprlz axa tsjrfukw lbv azqdsbmz aw zog iqlrl.

Npb egv gbmhluhlg nxfp axc qirnqq cpugu

(1,4)

Coordinate skills for area questions

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