Point $A (3, - 2)$ on reflection in the $X$ -axis is mapped as $A'$ ; and point $B$ on reflection in the $Y$ -axis is mapped onto $B' (- 4, 3)$ .
(i) Write down the co-ordinates of $A'$ and $B$ .
(ii) Find the slope of the line $A' B$ , hence find its inclination from $X$ -axis.

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Solution

Step1: Calculation of co-ordinates of point $A'$ .
The reflection of any point $(x, y)$ in the $X$ -axis is given by formula $(x, - y)$ .
For point $A (3, - 2)$ , $x = 3, y = - 2$ .
Thus, the reflection of point $A (3, - 2)$ in the $X$ -axis is given by:
$(3, - (- 2) = (3, 2)$
Hence, the co-ordinates of point $A'$ are $A' (3, 2)$ .
Step2: Calculation of co-ordinates of point $B$ .
The reflection of any point $(x, y)$ in the $Y$ -axis is given by formula $(- x, y)$ .
For point $B' (- 4, 3)$ , $x = - 4, y = 3$ .
Thus, the reflection of point $B' (- 4, 3)$ in the $Y$ -axis is given by:
$(- (- 4), 3) = (4, 3)$
Hence, the co-ordinates of point $B$ are $B (4, 3)$ .
Step3: Calculation of the slope of line $A' B$ .
The slope of a line passing through the points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is given by the formula $m = \frac{(y_{2} - y_{1})}{(x_{2} - x_{1})}$ .
For $A' (3, 2)$ and $B (4, 3)$ , $x_{1} = 3, y_{1} = 2, x_{2} = 4, y_{2} = 3$ .
Thus, the slope line $A' B$ is given by:
$m = \frac{3 - 2}{4 - 3} \Rightarrow m = \frac{1}{1} ∴ m = 1$
Thus, the slope of line $A' B$ is 1.
Step4: Calculation of the inclination of line $A' B$ with $X$ -axis.
The value of angle $θ$ made by a line having slope with the $X$ -axis is given by the formula $θ = \tan^{- 1} (m)$ .
Substitute the value of $m$ as 1 in the formula $θ = \tan^{- 1} (m)$ to obtain the inclination of line $A' B$ with $X$ -axis.
$θ = \tan^{- 1} (1) ∴ θ = 45 °$
Thus, the line $A' B$ makes an angle of $45 °$ with the $X$ -axis.
Hence,
(i) The coordinates of points $A'$ and $B$ are $A' (3, 2)$ and $B (4, 3)$ respectively.
(ii) The slope and the inclination of line $A' B$ from the $X$ -axis are 1 and $45 °$ respectively.

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Point A(3,-2)on reflection in the X-axis is mapped as A'; and point B on reflection in the Y-axis is mapped onto B'(-4,3).(i) Write down the co-ordinates of A' and B.(ii) Find the slope of the line A'B, hence find its inclination from X-axis.

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(i) Write down the co-ordinates of $A'$ and $B$ .
(ii) Find the slope of the line $A' B$ , hence find its inclination from $X$ -axis.