Question

Find the distance between \(A(-1,5)\) and \(C(3,5)\).

Short answer

The distance between A and C is 4 units.

Step-by-step solution

Identify Coordinates

Identify the coordinates of both points. The coordinates of point \(A\) are \((-1, 5)\) and the coordinates of point \(C\) are \((3, 5)\).

Check for Alignment

Notice that the y-coordinates are the same for both points (\(5\)), indicating that they are aligned horizontally.

Use Distance Formula for Horizontal Line

Since the points are on the same horizontal line, calculate the distance using the x-coordinates: the formula for distance between two points \((x_1, y)\) and \((x_2, y)\) is \(|x_2 - x_1|\).

Calculate the Distance

Subtract the x-coordinates: \(3 - (-1) = 3 + 1 = 4\). So, the distance between point \(A\) and point \(C\) is \(4\) units.

Key concepts

Coordinate Geometry

Coordinate geometry, often called analytical geometry, is a fascinating branch of mathematics that merges algebra and geometry. It involves describing geometric shapes and properties using a coordinate system. This powerful tool allows us to locate points uniquely on a plane using coordinates, typically denoted as \(x, y\).

These coordinates represent a point's position relative to two perpendicular lines or axes—called the x-axis (horizontal) and y-axis (vertical). Each point in the plane has a unique pair of coordinates. To identify a point entirely, we need both its x-coordinate and y-coordinate

• The x-coordinate tells us how far along the horizontal axis a point is.
• The y-coordinate indicates its position along the vertical axis.

The beauty of coordinate geometry is its ability to translate geometric problems into algebraic problems, which can often be simpler to solve.

Horizontal Alignment

Horizontal alignment is an essential concept when dealing with points on a plane using coordinate geometry. It refers to how points line up or align with respect to the horizontal axis.

In our specific exercise, recognizing that both points share the same y-coordinate (\(5\)) is critical. This indicates that points \(A\) and \(C\) are horizontally aligned. In simpler terms, they lie on a straight horizontal line parallel to the x-axis. When points have the same y-coordinate

• They are horizontally aligned.
• Their relative position along the y-axis remains constant.
• Only their x-coordinates vary.

Understanding horizontal alignment can significantly simplify the distance calculation between such points, as it reduces the problem to a single dimension along the x-axis.

Distance Calculation

Distance calculation in coordinate geometry often involves using the distance formula. However, when dealing with horizontally aligned points, the task gets more straightforward. The points \(A(-1, 5)\) and \(C(3, 5)\) have the same y-coordinate, simplifying our job.

Since we only need to calculate how far one point is from the other horizontally, we apply the simple formula \(|x_2 - x_1|\) for calculating the distance when points are on the same horizontal line.

• Identify the x-coordinates of both points. Here, they are -1 for point \(A\) and 3 for point \(C\).
• Calculate the absolute difference: \(3 - (-1) = 4\).

The distance is thus 4 units. This serves as a reminder of how coordinate geometry not only aids in solving complex geometric problems but also streamlines simple tasks like measuring lengths in a plane.