Question

Find the point if a a into (1 ,x) B (5, 2) c (9,5) if they are all collinear then find the value of x

Answer 1

Answer

The value of x is -1

Given

• The points A(1 , x) , B(5 , 2) and C(9 , 5)

To Find

• The value of x

Formula to be used

$\sf If \: A(x _{1} , y_{1}) \: and \: B(x_{2} ,y_{2}) \: are \: two \: points \\ \sf then \: their \: gradient \: or \: slop \: is \: given \\ \sf\: by :$

$\longrightarrow \boxed{ \boxed{ \tt Slope \: of \: line \: AB= \dfrac{ y_{2} - y_{1} }{x_{2} - x_{1}} }}$

Solution

The points are A(1 , x) , B(5 , 2) and C(9 , 5)

We know that when three points are collinear their slope or gradient are equal

$\sf \implies \dfrac{2-x}{5-1} = \dfrac{5-2}{9-5} \\\\ \sf \implies \dfrac{2-x}{4} = \dfrac{3}{4} \\\\ \sf\implies 4(2-x) = 4 \times 3 \\\\ \sf\implies 2-x = 3 \\\\ \sf\implies -x = 3-2 \\\\ \sf \implies -x = 1 \\\\ \sf\implies x = -1$

Answer 2

Given that ,

The three point are A(1,x) , B(5,2) , and C(9,5) are collinear

We know that , if three points P , Q and R are collinear then

$\large \sf\fbox{Slope \: of \: PQ = Slope \: of \: QR}$

Thus ,

$\sf \mapsto \frac{(2 - x)}{(5 - 1)} = \frac{(5 - 2)}{(9 - 5)} \: \: \: \{ \because slope \: (m) = \frac{ y_{2} - y_{1}}{x_{2} - x_{1}} \} \\ \\ \sf \mapsto \frac{(2 - x)}{4} = \frac{3}{4} \\ \\ \sf \mapsto 2 - x = 3 \\ \\ \sf \mapsto x = - 1$

$\sf \therefore \underline{The \: value \: of \: x \: is -1}$