Question

2. Determine Whether the points are collinear .R (0,3), D (2,1), S (3,-1) Maths part2 practice set 5.1 ​

Answer 1

$\large\underline\blue{\bold{Given \:  Question :-  }}$

• Determine whether the points are collinear :- R (0,3), D (2,1), S (3,-1)

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$\huge{AηsωeR} ✍$

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$\begin{gathered}\begin{gathered}\bf Given \: coordinates : - \begin{cases} &\sf{R (0,3)} \\ &\sf{D (2,1)}\\ &\sf{S (3,-1)} \end{cases}\end{gathered}\end{gathered}$

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$\begin{gathered}\begin{gathered}\bf To \: justify : - \begin{cases} &\sf{R, D, S \: are \: collinear \: or \: not.} \end{cases}\end{gathered}\end{gathered}$

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There are 4 Methods to justify whether points are collinear or not :-

$\begin{gathered}\begin{gathered}\bf Methods : - \begin{cases} &\sf{1. \: area \: of \: triangle} \\ &\sf{2. \: distance \: formula} \\ &\sf{3. \: section \: formula}\\ &\sf{4. \: slope \: method}\end{cases}\end{gathered}\end{gathered}$

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Prefer here, area of triangle

If the area of triangle formed by three points is zero, then they are said to be collinear.

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$\text{\large\underline{\orange{Formula:-}}}$

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$\sf \ Area =\frac{1}{2} [x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]$

$\begin{gathered}\begin{gathered}\bf where : - \begin{cases} &\sf{(x_{1} , y_{1}) \: is \: first \: vertex \: of \: triangle)} \\ &\sf{(x_{2} , y_{2} ) \: is \: the \: second \: vertex \: of \: triangle}\\ &\sf{(x_{3} , y_{3}) \: is \: third \: vertex \: of \: triangle )} \end{cases}\end{gathered}\end{gathered}$

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$\large\underline\purple{\bold{Solution :- }}$

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$\begin{gathered}\begin{gathered}\bf here : - \begin{cases} &\sf{(x_{1} , y_{1}) \: = (0,3)} \\ &\sf{(x_{2} , y_{2} ) \: = (2,1)}\\ &\sf{(x_{3} , y_{3}) \: (3, - 1) )} \end{cases}\end{gathered}\end{gathered}$

☆ Using formula,

$\sf \ Area =\frac{1}{2} [x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]$

$\sf \: Area = \frac{1}{2} [0(1 + 1) + 2( - 1 - 3) + 3(3 - 1)]$

$\sf \:  Area = \frac{1}{2} [0 - 8 + 6]$

$\sf \:  ⟼Area = \frac{1}{2} [ - 2] = | - 1| = 1 \: sq. \: units.$

$\sf \:  ⟼Area \: is \: not \: equals \: to \: 0$

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$\large{\boxed{\boxed{\bf{Hence, points \: are \: not \: collinear.}}}}$

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Answer 2

Step-by-step explanation:

Refer to this attachment