Finding the Reflection of a Point in a Horizontal Line

The problem asks us to find the reflection of a specific point, (5, -3), in a specific line, y = 3. Understanding how reflections work in coordinate geometry is key to solving this problem.

Understanding Reflection in a Line

When a point is reflected in a line, the line of reflection acts like a mirror. The reflected point is on the opposite side of the line from the original point, and the distance from the original point to the line is equal to the distance from the reflected point to the line. Also, the line segment connecting the original point and its reflection is perpendicular to the line of reflection.

Reflection in a Horizontal Line

For a reflection in a horizontal line given by the equation \(y = k\), the x-coordinate of the point remains unchanged. The y-coordinate changes such that the line \(y=k\) is exactly in the middle (the perpendicular bisector) of the original y-coordinate and the new y-coordinate.

Let the original point be \((x_1, y_1)\) and the line of reflection be \(y = k\). The reflected point will be \((x_2, y_2)\).

• The x-coordinate remains the same: \(x_2 = x_1\).
• The y-coordinate \(y_2\) is such that the midpoint of the segment connecting \((x_1, y_1)\) and \((x_1, y_2)\) lies on the line \(y = k\). The midpoint is \(\left(x_1, \frac{y_1 + y_2}{2}\right)\).
• So, \(\frac{y_1 + y_2}{2} = k\).
• Multiplying by 2, we get \(y_1 + y_2 = 2k\).
• Rearranging to find \(y_2\), we get \(y_2 = 2k - y_1\).

Thus, the formula for the reflection of a point \((x_1, y_1)\) in the horizontal line \(y = k\) is \((x_1, 2k - y_1)\).

Applying the Formula to Point (5, -3) and Line y = 3

In this problem, the original point is \((x_1, y_1) = (5, -3)\) and the line of reflection is \(y = 3\), so \(k = 3\).

Using the formula for the reflected point \((x_2, y_2)\):

• \(x_2 = x_1 = 5\)
• \(y_2 = 2k - y_1 = 2(3) - (-3)\)
• \(y_2 = 6 - (-3)\)
• \(y_2 = 6 + 3\)
• \(y_2 = 9\)

So, the reflected point is \((5, 9)\).

Verification of the Reflection

Let's check if the properties of reflection hold for the point (5, -3) and its reflection (5, 9) in the line y = 3.

• Both points have the same x-coordinate, 5. This means the line segment connecting them is vertical.
• The midpoint of the segment connecting (5, -3) and (5, 9) is \(\left(\frac{5+5}{2}, \frac{-3+9}{2}\right) = \left(\frac{10}{2}, \frac{6}{2}\right) = (5, 3)\).
• The midpoint (5, 3) lies on the line y = 3, since its y-coordinate is 3.
• The line segment connecting (5, -3) and (5, 9) is vertical (same x-coordinate), and the line y = 3 is horizontal. A vertical line is perpendicular to a horizontal line.

These checks confirm that (5, 9) is indeed the correct reflection of (5, -3) in the line y = 3.

Comparison with Options

Comparing our result (5, 9) with the given options:

The calculated reflection (5, 9) matches Option 3.

Revision Table: Reflection in Coordinate Geometry

Additional Information: Transformations in Geometry

Reflection is one type of geometric transformation. Other common transformations include translation, rotation, and dilation. Transformations change the position, size, or orientation of a geometric figure.

• Translation: Sliding a figure without changing its size, shape, or orientation. Defined by a vector \((a, b)\), translating \((x, y)\) gives \((x+a, y+b)\).
• Rotation: Turning a figure about a fixed point (the center of rotation) by a specific angle.
• Dilation: Resizing a figure by a scale factor from a fixed point (the center of dilation). A dilation centered at the origin with scale factor \(s\) transforms \((x, y)\) to \((sx, sy)\).

These transformations are fundamental concepts in coordinate geometry and are often studied together to understand how shapes move and change on a plane.