Question

Points P, Q, R and S, in that order, divide a line segment AB, into 5 equal parts. A is at ( 1, 2)  and B is at ( 6, 7 ). P is closest to A.Find the coordinates of S.​

Answer 1

$\large\text{\underline{Let's begin.}}$

$\large{\text{Basic Concepts}}$

$\text{Internal division formula.}$

$\hookrightarrow \large\boxed{P\left(\dfrac{mx_{2}+nx_{1}}{m+n}, \dfrac{my_{2}+ny_{1}}{m+n}\right)}$

$\text{Explanation for the formula.}$

Here, $(x_{1},y_{1}),(x_{2},y_{2})$ are two coordinates and $m:n$ is the ratio of division.

$\large\text{\underline{Solution}}$

$\overline{AB}$ is internally and equally divided by four points $P,Q,R,S$, in order. $P$ is the closest point to $A$.

$\hookrightarrow \overline{AP}:\overline{PQ}:\overline{QR}:\overline{RS}:\overline{SB}=1:1:1:1:1$

$\hookrightarrow \large\boxed{\overline{AS}:\overline{SB}=4:1}$

As S divides $\overline{AB}$ into $4:1$ internally, to apply the formula we find the values,

• $m=4,n=1$
• $x_{1}=1,x_{2}=6$
• $y_{1}=2,y_{2}=7$

Then,

$\hookrightarrow S\left(\dfrac{4\times6+1\times1}{4+1},\dfrac{4\times7+1\times2}{4+1}\right)$

$\hookrightarrow S(\dfrac{25}{5},\dfrac{30}{5})$

$\therefore \large\boxed{S(5,6)}$

$\large\text{\underline{Conclusion}}$

Hence, the coordinates of S are $(5,6)$.

$\large{\text{\underline{Learn more.}}}$

$\text{Internal division formula.}$

$\hookrightarrow \large\boxed{P\left(\dfrac{mx_{2}+nx_{1}}{m+n}, \dfrac{my_{2}+ny_{1}}{m+n}\right)}$

$\text{Explanation for the formula.}$

It is used when a point on a line divides a line segment.

$\text{External division formula.}$

$\hookrightarrow \large\boxed{P\left(\dfrac{mx_{2}-nx_{1}}{m-n}, \dfrac{my_{2}-ny_{1}}{m-n}\right)}$

$\text{Explanation for the formula.}$

It is used when a point outside a line divides a line segment.

$\text{Midpoint formula.}$

$\hookrightarrow \large\boxed{P\left(\dfrac{x_{1}+x_{2}}{2}, \dfrac{y_{1}+y_{2}}{2}\right)}$

$\text{Explanation for the formula.}$

It is a special case of internal division, as the ratio of division is $1:1$.

Answer 2

Answer:

‣ Coordinates of S $\mapsto\:{\boxed{\tt{(5,\:6)}}}$

Explanation:

Given info,

Points P, Q, R and S, in that order, divide a line segment AB, into 5 equal parts. A is at (1, 2) and B is at (6, 7). P is closest to A. Find the coordinates of S.

• Coordinates of A = (1, 2)
• Coordinates of B = (6, 7)
• Coordinates of S = ?

⚘ As points P, Q, R, S divides line segment into 5 equal parts. So,

• AP:PQ:QR:RS:SB = 1:1:1:1:1
• AS:SB = (1 + 1 + 1 + 1):1 = 4:1

⚘ Using section formula for finding coordinates of 'S' ::

‣ $\boxed{\underline{\underline{\bf{\red{\Bigg(\dfrac{m_{1}x_{2} + m_{2}x_{1}}{m_{1} + m _{2}},\:\dfrac{m_{1}y_{2} + m_{2}y_{1}}{m_{1} + m _{2}}\Bigg)}}}}}\:\bf{\dag}$

We have,

• $\sf x_{1}$ = 1, $\sf x_{2}$ = 6
• $\sf y_{1}$ = 2, $\sf y_{2}$ = 7
• $\sf m_{1}$ = 4, $\sf m_{2}$ = 1

Putting all values,

$\\ \longrightarrow\:\tt S = \Bigg(\dfrac{(4)(6) + (1)(1)}{4 + 1},\:\dfrac{(4)(7) + (1)(2)}{4 + 1}\Bigg)$

$\\ \longrightarrow\:\tt S = \Bigg(\dfrac{24 + 1}{5},\:\dfrac{28 + 2}{5}\Bigg)$

$\\ \longrightarrow\:\tt S = \Bigg({\cancel{\dfrac{25}{5}}},\:{\cancel{\dfrac{30}{5}}}\Bigg)$

$\\ \longrightarrow\:{\underline{\boxed{\tt{S = (5,\:6)}}}}\:\bigstar$

• Hence, coordinates of S are (5, 6).

More to know:

• For finding distance between two points, we use distance formula i.e,$\sf \sqrt{\bigg(x_{2} - x_{1}\bigg)^2 + \bigg(y_{2} - y_{1}\bigg)^2}$

• For finding the ratio in which a line segment is divided by a point, we use section formula i.e, $\sf \bigg(\dfrac{m_{1}x_{2} + m_{2}x_{1}}{m_{1} + m_{2}}\:,\:\dfrac{m_{1}y_{2} + m_{2}y_{1}}{m_{1} + m_{2}}\bigg)$

• The area of triangle formed by points $\sf (x_{1},\:y_{1})$, $\sf (x_{2},\:y_{2})$ and $\sf (x_{3},\:y_{3})$ is the numerical value of the expression :: $\small\sf\dfrac{1}{2}\Big[x_{1}\big(y_{2} - y_{3}\big) + x_{2}\big(y_{3} - y_{1}\big) + x_{3}\big(y_{1} - y_{2}\big)\Big]$

• The mid - point of the line segment joining the points $\sf A(x_{1},\:y_{1})$ and $\sf B(x_{2},\:y_{2})$ is :: $\sf \Bigg(\dfrac{x_{1} + x_{2}}{2},\:\dfrac{y_{1} + y_{2}}{2}\Bigg)$

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