Problem 20 - Algebra 2

Question

Find the distance between each pair of points with the given coordinates. $$ (0.23,0.4),(0.68,-0.2) $$

Step-by-Step Solution

Answer

The distance between the points is 0.75.

Step 1: Calculate the Change in x-coordinates (Δx)

Identify the x-coordinates of the two points, which are 0.23 and 0.68. Calculate the change in the x-coordinates: \[ \Delta x = 0.68 - 0.23 \] So, \( \Delta x = 0.45 \).

Step 2: Calculate the Change in y-coordinates (Δy)

Identify the y-coordinates of the two points, which are 0.4 and -0.2. Calculate the change in the y-coordinates: \[ \Delta y = -0.2 - 0.4 \] So, \( \Delta y = -0.6 \).

Step 3: Use the Distance Formula

Use the distance formula, which is: \[ d = \sqrt{(\Delta x)^2 + (\Delta y)^2} \] Substitute \( \Delta x = 0.45 \) and \( \Delta y = -0.6 \) into the formula: \[ d = \sqrt{(0.45)^2 + (-0.6)^2} \] This simplifies to: \[ d = \sqrt{0.2025 + 0.36} = \sqrt{0.5625} \].

Step 4: Compute the Distance

Calculate the square root of the resulting sum: \[ d = \sqrt{0.5625} \]The distance is: \( d = 0.75 \).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where algebra is used to describe geometric concepts. In this type of geometry, points are defined through coordinates on a grid. Think of it like mapping locations on a graph paper, where each point is typically described using two numbers
in a pair known as an ordered pair. The first number in the pair represents
the horizontal position (x-coordinate) and the second number indicates the vertical position (y-coordinate).

• X-Coordinates: They measure the distance along the horizontal axis, which is often called the x-axis.
• Y-Coordinates: They measure the distance along the vertical axis, known as the y-axis.

A coordinate plane is divided into four quadrants by these axes, and each point in the plane corresponds to a unique pair of coordinates. This is essential for understanding how to navigate and measure distances in a planar, two-dimensional space.

Distance Calculation

The distance calculation in coordinate geometry involves determining the straight-line distance between two points on a coordinate plane. This is done using the Distance Formula, a straightforward method derived from the Pythagorean Theorem.
The formula calculates the hypotenuse of a right-angled triangle, where the legs represent the
differences in x and y coordinates (horizontal and vertical distances, respectively).Distance Formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

• The terms \(x_2\) and \(x_1\) are the x-coordinates of the second and first points.
• The terms \(y_2\) and \(y_1\) are the y-coordinates of the second and first points.

By substituting the specific differences in coordinates, we can find the precise distance
between the two points. For instance, given points \((0.23,0.4)\) and \((0.68,-0.2)\),
the changes in coordinates are plugged into the formula to achieve the final result. This practical technique
allows one to measure distances accurately in various geometry problems.

Change in Coordinates

The concept of change in coordinates refers to how much the x or y values of a point differ from those of another point. This change is crucial for calculating distances as it forms the basis for determining
the direction and magnitude of movement between points.

• Change in X (\(\Delta x\)) is calculated by subtracting the x-coordinate of the first point from the x-coordinate of the second point.
• Change in Y (\(\Delta y\)) is obtained by subtracting the y-coordinate of the first point from the y-coordinate of the second point.

In our example, for the given points \((0.23, 0.4)\) and \((0.68, -0.2)\):

• The change in x (\(\Delta x\)) is \(0.68 - 0.23 = 0.45\).
• The change in y (\(\Delta y\)) is \(-0.2 - 0.4 = -0.6\).

Understanding these changes helps visualize how far and in what direction one point is from another on the coordinate plane.
This basic yet critical step is necessary for performing accurate distance calculations using the Distance Formula.