Problem 29
Question
Find the midpoint of each line segment with the given endpoints. $$(\sqrt{18},-4)\( and \)(\sqrt{2}, 4)$$
Step-by-Step Solution
Verified Answer
The midpoint of the line segment with endpoints \( (\sqrt{18},-4) \) and \( (\sqrt{2}, 4) \) is \( (\sqrt{5}, 0) \).
1Step 1: Identify the Coordinates
Identify the coordinates of the given points. Let's assign: \( x_{1} = \sqrt{18}, y_{1} = -4 \) (the first point) and \( x_{2} = \sqrt{2}, y_{2} = 4 \) (the second point)
2Step 2: Solve for Midpoint
Substitute the given values into the midpoint formula. The midpoint M will be calculated as follows: \( M = ( \frac{{\sqrt{18} + \sqrt{2}}}{2}, \frac{{-4 + 4}}{2}) \)
3Step 3: Simplify the Results
After substitution, simplify the result. The x-coordinate: \( \frac{{\sqrt{18} + \sqrt{2}}}{2} \) simplifies to \( \frac {\sqrt{20}}{2} \) or \( \sqrt{5} \). The y-coordinate: \( \frac{{-4 + 4}}{2} \) becomes 0.
Key Concepts
Coordinate GeometryLine Segment MidpointRadicals Simplification
Coordinate Geometry
In the realm of mathematics, coordinate geometry, also known as analytic geometry, serves as a bridge between algebra and geometry by using a coordinate system. This system translates geometric figures and relationships into algebraic equations and vice versa.
In the specific context of finding the midpoint of a line segment, coordinate geometry uses a straightforward algebraic form: the midpoint formula. This formula calculates the average of the x-coordinates and the y-coordinates of the endpoints of a line segment to determine the precise point that lies exactly halfway between them.
Coordinate geometry not only aids in determining midpoints but also is integral to a multitude of applications such as determining slope, distance between two points, and the equations of various shapes and lines, making it an essential element of spatial reasoning and a staple in algebra and geometry curricula.
In the specific context of finding the midpoint of a line segment, coordinate geometry uses a straightforward algebraic form: the midpoint formula. This formula calculates the average of the x-coordinates and the y-coordinates of the endpoints of a line segment to determine the precise point that lies exactly halfway between them.
Coordinate geometry not only aids in determining midpoints but also is integral to a multitude of applications such as determining slope, distance between two points, and the equations of various shapes and lines, making it an essential element of spatial reasoning and a staple in algebra and geometry curricula.
Line Segment Midpoint
When we consider a line segment in coordinate geometry, the midpoint is the point that divides the segment into two congruent segments. It is essentially the average position between the two endpoints.
To find a line segment's midpoint, you would use the midpoint formula. This formula is written as: \( M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}) \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the endpoints of the line segment. The midpoint \( M \) is thus a pair of these averages - one for the x-coordinate and one for the y-coordinate.
This method is especially useful in applications such as computer graphics, navigation, and various fields of engineering, where pinpointing the center or balancing point of a segment is necessary.
To find a line segment's midpoint, you would use the midpoint formula. This formula is written as: \( M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}) \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the endpoints of the line segment. The midpoint \( M \) is thus a pair of these averages - one for the x-coordinate and one for the y-coordinate.
This method is especially useful in applications such as computer graphics, navigation, and various fields of engineering, where pinpointing the center or balancing point of a segment is necessary.
Radicals Simplification
Simplifying radicals is an essential skill in algebra that often comes into play when dealing with coordinate geometry. A radical is an expression that includes a square root, cube root, or other higher root symbol.
To simplify a radical, we aim to find the largest square factor of the number under the radical and express it as a product of square roots. For example, with \( \sqrt{18} \), we can recognize that 18 is \( 9 \times 2 \), and since 9 is a perfect square, we can simplify \( \sqrt{18} \) to \( \sqrt{9} \times \sqrt{2} \), which is \( 3\sqrt{2} \).
This process reduces the expression to its simplest form, which simplifies the calculation process and makes it easier to understand and work with the numbers, particularly when they are part of a larger algebraic formula such as the midpoint formula.
To simplify a radical, we aim to find the largest square factor of the number under the radical and express it as a product of square roots. For example, with \( \sqrt{18} \), we can recognize that 18 is \( 9 \times 2 \), and since 9 is a perfect square, we can simplify \( \sqrt{18} \) to \( \sqrt{9} \times \sqrt{2} \), which is \( 3\sqrt{2} \).
This process reduces the expression to its simplest form, which simplifies the calculation process and makes it easier to understand and work with the numbers, particularly when they are part of a larger algebraic formula such as the midpoint formula.
Other exercises in this chapter
Problem 28
Find the domain of each function. $$ g(x)-\frac{\sqrt{x-3}}{x-6} $$
View solution Problem 28
Determine whether each function is even, odd, or neither. $$f(x)=x^{2} \sqrt{1-x^{2}}$$
View solution Problem 29
Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((-3,-1)\) and \((2,4)\)
View solution Problem 29
Find the domain of each function. $$ f(x)-\frac{2 x+7}{x^{3}-5 x^{2}-4 x+20} $$
View solution