Question

Find the distance between \((-2,3)\) and the midpoint of the segment joining \((-2,-2)\) and \((4,3)\).

Short answer

The distance is \(\frac{\sqrt{61}}{2}\).

Step-by-step solution

Find the Midpoint

To find the midpoint of the segment joining \((-2, -2)\) and \((4, 3)\), use the midpoint formula: \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). Substitute \(x_1 = -2\), \(y_1 = -2\), \(x_2 = 4\), \(y_2 = 3\). This results in: \(\left(\frac{-2 + 4}{2}, \frac{-2 + 3}{2}\right) = \left(1, \frac{1}{2}\right)\). The midpoint is \((1, \frac{1}{2})\).

Apply the Distance Formula

Now, find the distance between \((-2, 3)\) and the midpoint \((1, \frac{1}{2})\) using the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Here, \(x_1 = -2\), \(y_1 = 3\), \(x_2 = 1\), \(y_2 = \frac{1}{2}\).

Substitute into the Distance Formula

Substitute the values into the distance formula: \(d = \sqrt{(1 - (-2))^2 + \left(\frac{1}{2} - 3\right)^2}\). Simplify: \(d = \sqrt{(1 + 2)^2 + \left(-\frac{5}{2}\right)^2}\). This becomes \(d = \sqrt{3^2 + \left(-\frac{5}{2}\right)^2}\).

Calculate the Terms

Calculate the squares: \(3^2 = 9\) and \(\left(-\frac{5}{2}\right)^2 = \frac{25}{4}\). Combine them: \(d = \sqrt{9 + \frac{25}{4}}\).

Combine and Simplify

To combine the terms with a common denominator: \(9 = \frac{36}{4}\). So, \(d = \sqrt{\frac{36}{4} + \frac{25}{4}} = \sqrt{\frac{61}{4}}\).

Final Simplification

Simplify by taking the square root: \(d = \sqrt{\frac{61}{4}} = \frac{\sqrt{61}}{2}\).

Key concepts

Midpoint Formula

To understand the midpoint formula, imagine you have a straight line connecting two points on a coordinate plane. The midpoint is exactly halfway between these two points. It’s like finding the center point of a segment. The formula is \[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]In simpler terms:

• Add the x-coordinates of the two points together and then divide by 2 to find the x-value of the midpoint.
• Add the y-coordinates of the two points together and divide by 2 to find the y-value of the midpoint.

For example, if your two points are \((-2, -2)\) and \((4, 3)\), you're adding their x-values \(-2 + 4 = 2\) and y-values \(-2 + 3 = 1\).Divide by 2 to reach the midpoint \((1, \frac{1}{2})\).This formula helps you find a balanced point between two extremes.

Coordinate Geometry

Coordinate geometry, often called analytic geometry, is the study of geometry using a coordinate system. It is a powerful mathematical tool for describing and analyzing geometric shapes in a plane. Here are some essential ideas:

• Points are represented as coordinates \((x, y)\) on a plane.
• Lines can be described using algebraic equations.
• Shapes and curves can be analyzed using equations and distances between points.

This approach combines algebra and geometry, enabling us to calculate distances, angles, and other properties much more easily.For instance, finding the midpoint or using the distance formula both rely on coordinate geometry. In the exercise, both the original segment and the distance between points are expressed in precise locations on this coordinate system, allowing for accurate calculations.

Euclidean Distance

Euclidean distance is the "straight-line" distance between two points in Euclidean space, which is the basic two-dimensional plane used in geometry. When we say "distance," this is typically what we mean. It is calculated using the formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]In simple words:

• Subtract the x-coordinates and square the result.
• Repeat for the y-coordinates.
• Add these squares together.
• Take the square root of the sum to find the Euclidean distance.

In our exercise, you needed to find the distance between point \((-2, 3)\) and the midpoint \((1, \frac{1}{2})\). By plugging the coordinates into the formula, you calculate each term and combine them to find that distance is \(\frac{\sqrt{61}}{2}\).Euclidean distance is a fundamental concept in geometry and it is widely used in fields like physics, engineering, and computer science.