4.2 Q-2

Question Statement

Find the coordinates of point P in the original xy-coordinate system, given that the axes have been translated through point $O'$ . The coordinates of point $P$ in the translated system are provided.

Background and Explanation

This problem involves translating the coordinates of a point from a translated coordinate system (with origin at $O'$ ) back to the original xy-coordinate system. To do this, we need to reverse the translation by adding the coordinates of the new origin $O'$ to the coordinates of point $P$ . The general formula for translating back to the original system is:

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

Where:

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

Solution

i. For P(8, 10) and O(3, 4)

Here, we have:

$P(8, 10)$ , and
$O(3, 4)$ .

Now, apply the translation formula:

$x = x' + h = 8 + 3 = 11$ ,
$y = y' + k = 10 + 4 = 14$ .

Thus, the coordinates of point P in the original system are:

P(11, 14).

ii. For P(-5, -3) and O(3, 4)

Here, we have:

$P(-5, -3)$ , and
$O(3, 4)$ .

Now, apply the translation formula:

$x = x' + h = -5 + 3 = -2$ ,
$y = y' + k = -3 + 4 = 1$ .

Thus, the coordinates of point P in the original system are:

P(-2, 1).

iii. For P(-3/4, -7/6) and O’(1/4, -1/6)

Here, we have:

$P\left(\frac{-3}{4}, \frac{-7}{6}\right)$ , and
$O'\left(\frac{1}{4}, \frac{-1}{6}\right)$ .

Now, apply the translation formula:

$x = x' + h = \frac{-3}{4} + \frac{1}{4} = \frac{-2}{4} = \frac{-1}{2}$ ,
$y = y' + k = \frac{-7}{6} + \frac{-1}{6} = \frac{-8}{6} = \frac{-4}{3}$ .

Thus, the coordinates of point P in the original system are:

P\left(\frac{-1}{2}, \frac{-4}{3}\right).

iv. For P(4, -3) and O’(-2, 3)

Here, we have:

$P(4, -3)$ , and
$O'(-2, 3)$ .

Now, apply the translation formula:

$x = x' + h = 4 + (-2) = 2$ ,
$y = y' + k = -3 + 3 = 0$ .

Thus, the coordinates of point P in the original system are:

P(2, 0).

Key Formulas or Methods Used

Translation of Coordinates:

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

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

Summary of Steps

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