Reflection

 Reflection                practice sums👈

Reflection of a Point in Coordinate Geometry

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📘 What is Reflection?

Reflection in geometry means "flipping" a point over a line to get its mirror image.
In coordinate geometry, this is done over axes like the x-axis, y-axis, or the line x = y.

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🔹 Reflection Rules

Line of Reflection	Original Point (x, y)	Image after Reflection
x-axis	(x, y)	(x, -y)
y-axis	(x, y)	(-x, y)
origin (0, 0)	(x, y)	(-x, -y)
line x = y	(x, y)	(y, x)

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📝 Explanation with Examples

🔸 1. Reflection in x-axis:

Flip vertically (change sign of y)

Example:
Reflect (3, 4) → (3, -4)

🔸 2. Reflection in y-axis:

Flip horizontally (change sign of x)

Example:
Reflect (3, 4) → (-3, 4)

🔸 3. Reflection in the origin:

Change signs of both x and y

Example:
Reflect (3, 4) → (-3, -4)

🔸 4. Reflection in line x = y:

Swap x and y

Example:
Reflect (3, 4) → (4, 3)

 

Reflection in the Line $x = a$

📘 Concept:

The line $x = a$ is a vertical line on the coordinate plane.
To reflect a point across $x = a$, imagine flipping the point over this vertical line like a mirror.

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🧠 Rule to Remember:

If a point $P(x, y)$ is reflected in the vertical line $x = a$
then the image $P'(x', y')$will be:

$$P' = (2a - x,\ y)$$

✅ Only the x-coordinate changes, the y-coordinate stays the same.

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🧮 Example 1:

Reflect the point $(4, 3)$in the line $x = 2$

Using the formula:

$$x' = 2a - x = 2×2 - 4 = 0,\quad y' = 3$$

→ Image = $(0, 3)$

Reflection in the Line $y=b$

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📘 Concept:

The line $y = b$ is a horizontal line on the coordinate plane.

To reflect a point across $y = b$, you flip it over this horizontal line — like a mirror placed horizontally at $y = b$.

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🧠 Rule to Remember:

If a point $P(x, y)$ is reflected in the horizontal line $y = b$,
then the image $P'(x', y')$ will be:

$$P' = (x,\ 2b - y)$$

✅ Only the y-coordinate changes, the x-coordinate stays the same.

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🧮 Example 1:

Reflect the point $(2, 5)$ in the line $y = 3$

$$y' = 2×3 - 5 = 6 - 5 = 1,\quad x' = 2$$

→ Image = $(2, 1)$