4.2 Q-1

Question Statement

Find the coordinates of point P referred to the translated axes $O'X$ and $O'Y$ , given the points in the xy-coordinate system.

Background and Explanation

This problem involves translating a point from one coordinate system to another. To do so, we need to subtract the origin coordinates of the translated axes from the original coordinates of point P. The general formula for translating the point is:

x' = x - h, \quad y' = y - k

Where:

$(x, y)$ are the original coordinates of the point P,
$(h, k)$ are the coordinates of the translated origin point $O'$ ,
$(x', y')$ are the new coordinates of point P in the translated system.

Solution

i. For P(3, 2) and O’(1, 3)

Here, we have:

$P(3, 2)$ , and
$O'(1, 3)$ .

Now, apply the translation formulas:

$x' = x - h = 3 - 1 = 2$ ,
$y' = y - k = 2 - 3 = -1$ .

Thus, the new coordinates of P are:

P'(2, -1).

ii. For P(-2, 6) and O’(-3, 2)

Here, we have:

$P(-2, 6)$ , and
$O'(-3, 2)$ .

Now, apply the translation formulas:

$x' = x - h = -2 - (-3) = -2 + 3 = 1$ ,
$y' = y - k = 6 - 2 = 4$ .

Thus, the new coordinates of P are:

P'(1, 4).

iii. For P(-6, -8) and O’(-4, -6)

Here, we have:

$P(-6, -8)$ , and
$O'(-4, -6)$ .

Now, apply the translation formulas:

$x' = x - h = -6 - (-4) = -6 + 4 = -2$ ,
$y' = y - k = -8 - (-6) = -8 + 6 = -2$ .

Thus, the new coordinates of P are:

P'(-2, -2).

iv. For P(3/2, 5/2) and O’(-1/2, 7/2)

Here, we have:

$P\left(\frac{3}{2}, \frac{5}{2}\right)$ , and
$O'\left(\frac{-1}{2}, \frac{7}{2}\right)$ .

Now, apply the translation formulas:

$x' = x - h = \frac{3}{2} - \frac{-1}{2} = \frac{3}{2} + \frac{1}{2} = 2$ ,
$y' = y - k = \frac{5}{2} - \frac{7}{2} = \frac{5 - 7}{2} = \frac{-2}{2} = -1$ .

Thus, the new coordinates of P are:

P'(2, -1).

Key Formulas or Methods Used

Translation of Coordinates:

x' = x - h, \quad y' = y - k

Where $(x, y)$ are the original coordinates of point P, and $(h, k)$ are the coordinates of the translated origin point $O'$ .

Summary of Steps

Identify the coordinates of point P and the translated origin $O'$ .
Use the translation formula $x' = x - h$ and $y' = y - k$ to find the new coordinates.
Repeat for all given points and their respective origins.