Problem Setup: 2D Point and Reflection Line
We need to find the mirror image of the 2D point P(5, 10) after reflection across a line that passes through the origin (0,0) and makes an angle of 45° counterclockwise with the positive X-axis.
Given:
- Point coordinates: $(x, y) = (5, 10)$
- Line angle with X-axis: $\theta = 45^\circ$
Reflection Formula for a Line Through Origin
The coordinates $(x', y')$ of a point $(x, y)$ reflected across a line passing through the origin with an angle $\theta$ to the positive X-axis are given by the formulas:
- $x' = x \cos(2\theta) + y \sin(2\theta)$
- $y' = x \sin(2\theta) - y \cos(2\theta)$
Calculate Angle and Trigonometric Values
First, determine the value of $2\theta$ and its cosine and sine:
- $2\theta = 2 \times 45^\circ = 90^\circ$
- $\cos(2\theta) = \cos(90^\circ) = 0$
- $\sin(2\theta) = \sin(90^\circ) = 1$
Calculate Transformed Coordinates
Substitute the values of $(x, y)$, $\cos(2\theta)$, and $\sin(2\theta)$ into the reflection formulas:
- $x' = (5) \times \cos(90^\circ) + (10) \times \sin(90^\circ)$
$x' = 5 \times 0 + 10 \times 1$
$x' = 0 + 10 = 10$ - $y' = (5) \times \sin(90^\circ) - (10) \times \cos(90^\circ)$
$y' = 5 \times 1 - 10 \times 0$
$y' = 5 - 0 = 5$
Final Transformed Point
The coordinates of the mirror image point after reflection are (10, 5).
Important Questions from Mirroring of Shapes
Examples of mirror and water reflections are shown in the figures below:
An object appears as the following image after first reflecting in a mirror and then reflecting on water.
The original object is
A line of symmetry is defined as a line that divides a figure into two parts in a way such that each part is a mirror image of the other part about that line.
The figure below consists of 20 unit squares arranged as shown. In addition to the given black squares, upto 5 more may be coloured black. Which one among the following options depicts the minimum number of boxes that must be coloured black to achieve two lines of symmetry? (The figure is representative)
The mirror image of the above text about the X-axis is
The least number of squares that must be added so that the line P-Q becomes the line of symmetry is ___________
For the picture shown above, which one of the following is the correct picture representing reflection with respect to the mirror shown as the dotted line?
