# Points A and B are at (8 ,2 ) and (1 ,7 ), respectively. Point A is rotated counterclockwise about the origin by pi/2  and dilated about point C by a factor of 3 . If point A is now at point B, what are the coordinates of point C?

Apr 7, 2016

color(green)("Point C "-> P_C ->(x,y)->(-3 1/2" "," "8 1/2))

#### Explanation:

$\text{Point "A_1" is rotated through " pi/2" to point } {A}_{2}$

Points $C \text{ "A_2" and } B$ form a straight line

Distance C to B is 3 times the distance C to ${A}_{2}$

$\textcolor{red}{\text{Solved using ratios of triangle sides}}$
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$\textcolor{b l u e}{\text{Determine } {x}_{C}}$

$\textcolor{b r o w n}{\text{Taking us along the x-axis from B to } {A}_{2}}$
$\implies {x}_{{A}_{2}} = {x}_{B} - \left({x}_{B} - {x}_{A}\right)$

$\textcolor{b r o w n}{\text{Taking us along the x-axis from B to C}}$

But from B to C is $\frac{1}{2}$ as much again giving us the 3 halves.

$\implies {x}_{C} = {x}_{B} - \left({x}_{B} - {x}_{A}\right) - \left(\frac{{x}_{B} - {x}_{A}}{2}\right)$

$\textcolor{b l u e}{\implies {x}_{C} = 1 - \left(3\right) - \left(1 \frac{1}{2}\right) = - 3 \frac{1}{3}}$

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$\textcolor{b l u e}{\text{Determine } {y}_{C}}$
$\textcolor{b r o w n}{\text{Taking us along the y-axis from B to } {A}_{2}}$

$\implies {y}_{{A}_{2}} = {y}_{B} + \left({y}_{{A}_{2}} - {y}_{B}\right)$

$\textcolor{b r o w n}{\text{Taking us along the y-axis from B to C}}$

But from B to C is $\frac{1}{2}$ as much again giving us the 3 halves.

$\implies {y}_{{A}_{2}} = {y}_{B} + \left({y}_{{A}_{2}} - {y}_{B}\right) + \frac{{y}_{{A}_{2}} - {y}_{B}}{2}$

$\textcolor{b l u e}{\implies {y}_{{A}_{2}} = 7 + \left(1\right) + \left(\frac{1}{2}\right) = 8 \frac{1}{2}}$
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color(green)("So point C "-> P_C ->(x,y)->(-3 1/2" "," "8 1/2))