Problem 34 - Algebra and Trigonometry

Question

Which of the points \(C(-6,3)\) or \(D(3,0)\) is closer to the point \(E(-2,1) ?\)

Step-by-Step Solution

Answer

Point C is closer to point E.

Step 1: Understanding Distance Formula

The distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in a coordinate plane is given by the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]We need this formula to find the distances from E to both C and D.

Step 2: Calculating Distance from C to E

Substitute the coordinates for points C \( (-6, 3) \) and E \( (-2, 1) \) into the distance formula:\[ d_{CE} = \sqrt{((-2) - (-6))^2 + (1 - 3)^2} \]This simplifies to:\[ d_{CE} = \sqrt{4^2 + (-2)^2} \]Further simplification gives:\[ d_{CE} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \]

Step 3: Calculating Distance from D to E

Next, substitute the coordinates for points D \( (3, 0) \) and E \( (-2, 1) \) into the distance formula:\[ d_{DE} = \sqrt{((3) - (-2))^2 + (0 - 1)^2} \]This simplifies to:\[ d_{DE} = \sqrt{5^2 + (-1)^2} \]Further simplification gives:\[ d_{DE} = \sqrt{25 + 1} = \sqrt{26} \]

Step 4: Comparing the Distances

Now compare the two distances calculated:- The distance from C to E is \( 2\sqrt{5} \).- The distance from D to E is \( \sqrt{26} \).Since \( 2\sqrt{5} \approx 4.47 \) and \( \sqrt{26} \approx 5.10 \, \) point C is closer to point E than point D.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional space defined by an x-axis (horizontal) and y-axis (vertical). This system allows us to pinpoint exact locations using pairs of numbers, called coordinates.

Understanding how a coordinate plane works is essential to solving problems that involve distances between points. Each point is represented as often written in the form \((x, y)\). For example, the location of point \(C\) is \((-6, 3)\) on the plane. This means \(-6\) is the value on the x-axis, and \(3\) is the value on the y-axis.

Through this grid-like system:

• We can plot multiple points, such as \(D(3,0)\) and \(E(-2,1)\).
• We can visually perceive the spatial relationships between these points.
• It sets the foundation for calculating distances.

Distance Calculation

To find the distance between two points on a coordinate plane, we use the Distance Formula. This formula is derived from the Pythagorean Theorem and helps us calculate the straight-line distance (or Euclidean distance) between any two points:

• The formula is:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

In our example, we need to calculate distances from point \(E(-2,1)\) to both points \(C(-6,3)\) and \(D(3,0)\). Each calculation follows the same methodology:

• Subtract the coordinates of one point from those of the other.
• Square the results.
• Add the squared values.
• Take the square root of this sum to find the distance.

The step-by-step simplification results in:

• \(d_{CE} = \sqrt{20} = 2\sqrt{5}\)
• \(d_{DE} = \sqrt{26}\)

This method ensures that we always measure the true distance, ignoring any bends or curves in the path.

Comparison of Distances

After calculating distances, comparing them helps determine which point is closer to the reference point. Using the derived distances: \(d_{CE}\) and \(d_{DE}\), we can assess proximity.

If we followed the previous calculations:

The distance from \(C\) to \(E\) is \(2\sqrt{5} \approx 4.47\).

The distance from \(D\) to \(E\) is \(\sqrt{26} \approx 5.10\).

Through comparison:

• Point \(C(-6,3)\) is closer to \(E(-2,1)\), compared to point \(D(3,0)\).

This practical realization shows how effective the Distance Formula is in real situations where exact measurements matter.