Problem 27 - College Algebra Essentials

Question

find the midpoint of each line segment with the given endpoints. $$ (8,3 \sqrt{5}) \text { and }(-6,7 \sqrt{5}) $$

Step-by-Step Solution

Answer

The midpoint of the line segment with endpoints \((8,3\sqrt{5})\) and \((-6,7\sqrt{5})\) is \((1,5\sqrt{5})\).

Step 1: Identify the coordinates

First, identify the coordinates of the endpoints. In this case, the coordinates of one endpoint are \((8,3\sqrt{5})\) and the other are \((-6,7\sqrt{5})\). So, \(x_1=8, y_1=3\sqrt{5}, x_2=-6, y_2=7\sqrt{5}\).

Step 2: Apply the midpoint formula

Next, apply the midpoint formula. The x-coordinate of the midpoint is given by \(\frac{x_1+x_2}{2}\) and the y-coordinate is given by \(\frac{y_1+y_2}{2}\). For the x-coordinate, substitute the values into the formula to get \(\frac{8+(-6)}{2}\). For the y-coordinate, substitute to get \(\frac{3\sqrt{5}+7\sqrt{5}}{2}\).

Step 3: Compute the midpoints

The calculations will yield: the x-coordinate \(\frac{8+(-6)}{2}=\frac{2}{2}=1\), and the y-coordinate \(\frac{3\sqrt{5}+7\sqrt{5}}{2}=\frac{10\sqrt{5}}{2}=5\sqrt{5}\). Thus the coordinates of the midpoint are \((1,5\sqrt{5})\).

Key Concepts

Coordinate Geometry

In the realm of mathematics, coordinate geometry, also known as analytic geometry, is a powerful tool that connects algebra with geometric shapes. It allows us to analyze shapes, lines, and curves by translating them into algebraic equations using a coordinate system.

In coordinate geometry, the position of any point in the plane is determined by an ordered pair of numbers, known as coordinates. The most common system used is the Cartesian coordinate system, where the horizontal axis is labeled as the x-axis and the vertical axis as the y-axis. Each point in the plane is defined by an x (horizontal) and a y (vertical) coordinate.For example, locating the midpoint of a line segment involves finding a point that is exactly halfway between its endpoints. To find this midpoint, we use the midpoint formula, which requires the coordinates of the segment's endpoints.

Algebraic Expressions

Algebra plays a significant role in simplifying and solving geometrical problems by using algebraic expressions. An algebraic expression is a collection of numbers, variables, and operators (such as +, −, ×, ÷) that represents a mathematical object or relationship.

The beauty of algebra is in its ability to generalize mathematical concepts, such as the midpoint of a line segment between two points. In the case of the midpoint formula, we express it as \((x_{midpoint}, y_{midpoint}) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\),where the subscripts refer to the coordinates of the endpoints. Algebraic expressions help us derive a universal approach to solve problems that would otherwise require a different strategy for each specific case.

Square Roots

Square roots are a fundamental concept in algebra and are crucial for solving equations involving squares. The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 25, denoted as \(\sqrt{25}\), is 5 because \(5 \times 5 = 25\).

In geometry, square roots often appear in the context of the Pythagorean theorem and distances between points. In the exercise above, the term \(\sqrt{5}\) arises when dealing with irrational numbers, which cannot be expressed as simple fractions. Algebra allows us to work with square roots in a more symbolic form, thus enabling the manipulation of expressions that contain them, such as \(3\sqrt{5}\) or \(7\sqrt{5}\) without having to approximate their decimal values.