The equation of the line AB is given
by y - y1 = m(x - x1), where
m is the gradient and x1 and y1 are the
coordinates of one of the points.
$$ \hspace*{2 mm}\mathrm{1.1\kern3mmm(AB)\ =\ \frac{y_B\ -\ y_A}{x_B\ -\ x_A}\kern2mm\ } $$
$$ \hspace*{22 mm}\mathrm{=\ \frac{12 − 2}{8 − 3}\ \kern4mm\ =\ \frac{10}{5}\kern2mm\ } $$
$$ \hspace*{22 mm}\mathrm{=\ 2\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{Now\ use\ the\ formula\ to\ determine\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{the\ equation\ of\ the\ line\ by\ keying\ in\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{the\ coordinates\ of\ one\ of\ the\ points\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{At\ point\ A(3 ; 3)\ :\ y - 2 = 2(x - 3)\kern2mm\ } $$
$$ \hspace*{38 mm}\mathrm{y - 2 = 2x - 6\kern2mm\ } $$
$$ \hspace*{38 mm}\mathrm{y = 2x - 4\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{The\ equation\ is\ :\ y = 2x - 4\kern2mm\ } $$
$$ \hspace*{2 mm}\mathrm{1.2\kern3mmm(CD)\ =\ \frac{y_D\ -\ y_C}{x_D\ -\ x_C}\kern2mm\ } $$
$$ \hspace*{22 mm}\mathrm{=\ \frac{9 − 7}{5 − 2}\kern2mm\ } $$
$$ \hspace*{22 mm}\mathrm{=\ \frac{2}{3}\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{At\ point\ C(2 ; 7)\ :\ y - 7 = \frac{2}{3}(x - 2)\kern2mm\ } $$
$$ \hspace*{38 mm}\mathrm{y - 7 = \frac{2x}{3} - \frac{4}{3}\kern2mm\ } $$
$$ \hspace*{34 mm}\mathrm{3y - 21 = 2x - 4\ \ \ \ \ \times 3\kern2mm\ } $$
$$ \hspace*{41 mm}\mathrm{3y = 2x + 17\ \ \ \ \ \times 3\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{2x - 3y + 17 = 0\kern2mm\ } $$
$$ \hspace*{34 mm}\mathrm{\bold{or}\ \kern3mm\ y = \frac{2}{3}x + \frac{17}{3}\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{The\ equation\ is\ :\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{3y = 2x + 17\ \ or\ 2x - 3y + 17 = 0\ \ or\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{y = \frac{2}{3}x + \frac{17}{3}\kern2mm\ } $$
$$ \hspace*{2 mm}\mathrm{1.3\kern3mmm(KL)\ =\ \frac{y_L\ -\ y_K}{x_L\ -\ x_K}\kern2mm\ } $$
$$ \hspace*{22 mm}\mathrm{=\ \frac{−1 − (−5)}{−3 − (−8)}\kern2mm\ } $$
$$ \hspace*{22 mm}\mathrm{=\ \frac {4}{5}\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{At\ point\ L(−3 ; −1)\ :\ y - (−1) = \frac{4}{5}(x - (−3))\kern2mm\ } $$
$$ \hspace*{46 mm}\mathrm{y + 1 = \frac{4x}{5} + \frac{12}{5}\kern2mm\ } $$
$$ \hspace*{44 mm}\mathrm{5y + 5 = 4x + 12\ \ \ \ \ \times 5\kern2mm\ } $$
$$ \hspace*{49 mm}\mathrm{5y = 4x + 7\kern2mm\ } $$
$$ \hspace*{36 mm}\mathrm{4x - 5y + 7 = 0\kern2mm\ } $$
$$ \hspace*{42 mm}\mathrm{\bold{or}\ \kern3mm\ y = \frac{4}{5}x + \frac{12}{5}\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{The\ equation\ is\ :\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{5y = 4x + 7\ \ or\ 4x - 5y + 7 = 0\ \ or\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{y = \frac{4}{5}x + \frac{12}{5}\kern2mm\ } $$
$$ \hspace*{2 mm}\mathrm{1.4\kern3mmm(PQ)\ =\ \frac{y_Q\ -\ y_P}{x_Q\ -\ x_P}\kern2mm\ } $$
$$ \hspace*{22 mm}\mathrm{=\ \frac{−3 − 7}{1 − (−6)}\kern2mm\ } $$
$$ \hspace*{22 mm}\mathrm{=\ \frac {−10}{7}\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{At\ point\ Q(1 ; −3)\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{y - (−3) = \frac{−10}{7}(x - 1)\kern2mm\ } $$
$$ \hspace*{29 mm}\mathrm{y + 3 = \frac{−10x}{7} + \frac{10}{7}\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{7y + 21 = −10x + 10\ \ \ \ \ \times 7\kern2mm\ } $$
$$ \hspace*{32mm}\mathrm{7y = −10x - 11\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{10x + 7y + 11 = 0\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{\bold{or}\ \kern3mm\ y = \frac{−10}{7}x − \frac{11}{7}\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{The\ equation\ is\ :\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{5y = 4x + 7\ \ or\ 4x - 5y + 7 = 0\ \ or\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{y = \frac{−10}{7}x − \frac{11}{7}\kern2mm\ } $$
$$ \hspace*{2 mm}\mathrm{1.5\kern3mmm(RS)\ =\ \frac{y_S\ -\ y_R}{x_S\ -\ x_R}\kern2mm\ } $$
$$ \hspace*{22 mm}\mathrm{=\ \frac{−7 − (−12)}{1 − (−3)}\kern2mm\ } $$
$$ \hspace*{22 mm}\mathrm{=\ \frac {5}{4}\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{At\ point\ R(−3 ; −12)\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{y - (−12) = \frac{5}{4}(x - (−3))\kern2mm\ } $$
$$ \hspace*{29 mm}\mathrm{y + 12 = \frac{5x}{4} + \frac{15}{4}\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{4y + 48 = 5x + 15\ \ \ \ \ \times 4\kern2mm\ } $$
$$ \hspace*{33 mm}\mathrm{4y = 5x - 33\kern2mm\ } $$
$$ \hspace*{18 mm}\mathrm{5x − 4y − 33 = 0\kern2mm\ } $$
$$ \hspace*{27 mm}\mathrm{\bold{or}\ \kern3mm\ y = \frac{5}{4}x − \frac{33}{4}\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{The\ equation\ is\ :\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{4y = 5x − 33\ \ or\ 5x - 4y − 33 = 0\ \ or\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{y = \frac{5}{4}x − \frac{33}{4}\kern2mm\ } $$
$$ \hspace*{2 mm}\mathrm{1.6\kern3mmm(AB)\ =\ \frac{y_B\ -\ y_A}{x_B\ -\ x_A}\kern2mm\ } $$
$$ \hspace*{22 mm}\mathrm{=\ \frac{−4 − (−8)}{3 − (−6)}\kern2mm\ } $$
$$ \hspace*{22 mm}\mathrm{=\ \frac {4}{9}\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{At\ point\ B(3 ; −4)\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{y - (−4) = \frac{4}{9}(x - 3)\kern2mm\ } $$
$$ \hspace*{29 mm}\mathrm{y + 4 = \frac{4x}{9} + \frac{4}{3}\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{9y + 36 = 4x + 12\ \ \ \ \ \times 9\kern2mm\ } $$
$$ \hspace*{33 mm}\mathrm{9y = 4x - 24\kern2mm\ } $$
$$ \hspace*{18 mm}\mathrm{4x − 9y − 24 = 0\kern2mm\ } $$
$$ \hspace*{27 mm}\mathrm{\bold{or}\ \kern3mm\ y = \frac{4}{9}x − \frac{8}{3}\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{The\ equation\ is\ :\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{9y = 4x − 24\ \ or\ 4x - 9y − 24 = 0\ \ or\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{y = \frac{4}{9}x − \frac{8}{3}\kern2mm\ } $$
$$ \hspace*{2 mm}\mathrm{2.1\kern3mmm(AB)\ =\ 3\ and\ A(4 ; 9)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{Now\ use\ the\ formula\ to\ determine\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{the\ equation\ of\ the\ line\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{At\ point\ A(4 ; 9)\ :\ y - 9 = 3(x - 4)\kern2mm\ } $$
$$ \hspace*{38 mm}\mathrm{y - 9 = 3x - 12\kern2mm\ } $$
$$ \hspace*{38 mm}\mathrm{y = 3x - 3\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{The\ equation\ is\ :\ y = 3x - 3\kern2mm\ } $$
$$ \hspace*{2 mm}\mathrm{2.2\kern3mmFind\ the\ gradient\ of\ the\ line\ and\ thus\ the\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{the\ gradient\ of\ the\ line\ through\ B\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{line\ :\ y = −2x + 5\ and\ gradient\ is\ −2\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{Gradient\ of\ iine\ through\ B\ is\ m(\|) = −2\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{Parallel\ lines\ have\ equal\ gradients\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{Now\ use\ the\ formula\ to\ determine\ the\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{equation\ of\ the\ line\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{At\ point\ B(−4 ; 7)\ :\ y - 7 = −2(x - (−4))\kern2mm\ } $$
$$ \hspace*{41 mm}\mathrm{y - 7 = −2x - 8\kern2mm\ } $$
$$ \hspace*{47 mm}\mathrm{y = −2x - 1\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{The\ equation\ is\ :\ y = −2x - 1\kern2mm\ } $$
$$ \hspace*{2 mm}\mathrm{2.3\kern3mmLine\ 3x − 4y − 6 = 0\kern2mm\ } $$
$$ \hspace*{38 mm}\mathrm{4y = 3x − 6\kern2mm\ } $$
$$ \hspace*{39 mm}\mathrm{y = \frac{3}{4}x − \frac{3}{2}\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{m(line) = \frac{3}{4}\ \ \ and\ thus\ m(line\ through\ C\ =\ \frac{3}{4}\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{At\ point\ C(−4 ; −5) \ :\ \kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{y - (−5) = \frac{3}{4}(x - (−4))\kern2mm\ } $$
$$ \hspace*{29 mm}\mathrm{y + 5 = \frac{3}{4}x + 3\kern2mm\ } $$
$$ \hspace*{34 mm}\mathrm{y = \frac{3}{4}x - 2\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{The\ equation\ is\ :\ y = \frac{3}{4}x - 2\kern2mm\ } $$
$$ \hspace*{2 mm}\mathrm{2.4\kern3mmLine\ 2x + 3y − 9 = 0\kern2mm\ } $$
$$ \hspace*{38 mm}\mathrm{3y = −2x + 9\kern2mm\ } $$
$$ \hspace*{39 mm}\mathrm{y = −\frac{2}{3}x + 3\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{m(line) = \frac{−2}{3}\ \ \ and\ thus\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{m(line\ through\ D\ =\ −\frac{2}{3}\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{At\ point\ D(3 ; −5)\ :\ \kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{y - (−5) = −\frac{2}{3}(x - 3)\kern2mm\ } $$
$$ \hspace*{29 mm}\mathrm{y + 5 = −\frac{2}{3}x + 2\kern2mm\ } $$
$$ \hspace*{34 mm}\mathrm{y = −\frac{2}{3}x - 3\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{The\ equation\ is\ :\ y = −\frac{2}{3}x - 3\kern2mm\ } $$
$$ \hspace*{2 mm}\mathrm{2.5\kern3mmLine\ is\ y = 4x - 3\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{m(line) = 4\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{m(line) × m(perpendicular\ line) = −1\kern2mm\ } $$
$$ \hspace*{17 mm}\mathrm{4 × m(perpendicular\ line) = −1\kern2mm\ } $$
$$ \hspace*{38 mm}\mathrm{m(\perp)\ =\ −\frac{1}{4}\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{At\ pointBy\ punt\ E(4 ; 6)\ :\ \kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{y - 6 = −\frac{1}{4}(x − 4)\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{y − 6 = −\frac{1}{4}x + 1\kern2mm\ } $$
$$ \hspace*{30 mm}\mathrm{y = −\frac{1}{4}x + 7\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{The\ equation\ is\ :\ y = −\frac{1}{4}x + 7\kern2mm\ } $$
$$ \hspace*{2 mm}\mathrm{2.6\kern3mmLine\ is\ 3x + 2y − 9 = 0\kern2mm\ } $$
$$ \hspace*{38 mm}\mathrm{2y = −3x + 9\kern2mm\ } $$
$$ \hspace*{39 mm}\mathrm{y = −\frac{3}{2}x + \frac{9}{2}\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{m(line) = \frac{−3}{2}\ \ \ and\ thus\ m(\perp) × \frac{−3}{2} =\ −1\kern2mm\ } $$
$$ \hspace*{51 mm}\mathrm{m(\perp) =\ \frac{2}{3}\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{At\ point\ F(−5 ; −8)\ :\ \kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{y - (−8) = \frac{2}{3}(x − (−5))\kern2mm\ } $$
$$ \hspace*{29 mm}\mathrm{y + 8 = \frac{2}{3}x + \frac{10}{3}\kern2mm\ } $$
$$ \hspace*{33 mm}\mathrm{y = \frac{2}{3}x − \frac{14}{3}\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{The\ equation\ is\ :\ y = \frac{2}{3}x − \frac{14}{3}\kern2mm\ } $$