Problem 27
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
Verified Answer
The midpoint of the line segment with endpoints (8, 3\sqrt{5}) and (-6, 7\sqrt{5}) is (1, 5\sqrt{5}).
1Step 1: Identify the given coordinates
The given coordinates are (8, 3\sqrt{5}) and (-6, 7\sqrt{5}). So, the coordinates of the first point (x1, y1) are (8, 3\sqrt{5}), and the coordinates of the second point (x2, y2) are (-6, 7\sqrt{5}).
2Step 2: Plug the coordinates into the midpoint formula
The midpoint formula is ((x1 + x2)/2, (y1 + y2)/2). So, plug the given coordinates into the formula: ((8 + -6)/2, (3\sqrt{5} + 7\sqrt{5})/2).
3Step 3: Simplify the result
After calculation, the midpoint will be: ((8 - 6)/2, (3\sqrt{5} + 7\sqrt{5})/2) = (1, 5\sqrt{5}).
Key Concepts
Coordinate GeometryLine SegmentsDistance and Midpoint
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, involves using a coordinate system to investigate geometric problems. By placing points on a plane using numerical coordinates, we can solve equations related to distance and location. This approach allows us to bridge the gap between algebra and geometry, making it easier to calculate and visualize geometric concepts.
In coordinate geometry, each point is defined by an ordered pair \(x, y\), representing its location on the Cartesian plane. This facilitates the application of algebraic techniques to geometric problems, such as finding midpoints and distances between points.
In coordinate geometry, each point is defined by an ordered pair \(x, y\), representing its location on the Cartesian plane. This facilitates the application of algebraic techniques to geometric problems, such as finding midpoints and distances between points.
Line Segments
Line segments are an essential part of geometry, representing a part of a line constrained by two endpoints. Unlike a line, which extends infinitely in both directions, a line segment has a definite beginning and end. The endpoints provide the framework for various calculations, including finding the midpoint.
When analyzing line segments in coordinate geometry, each endpoint is associated with coordinates. This allows for straightforward application of formulas like the distance formula and the midpoint formula. By understanding the position of a line segment in the Cartesian plane, you can solve diverse problems related to shape and size.
When analyzing line segments in coordinate geometry, each endpoint is associated with coordinates. This allows for straightforward application of formulas like the distance formula and the midpoint formula. By understanding the position of a line segment in the Cartesian plane, you can solve diverse problems related to shape and size.
Distance and Midpoint
The concepts of distance and midpoint are vital in coordinate geometry. They help in understanding the relative positioning of points on a plane and in finding central points efficiently.
The **midpoint formula** is specifically used to find the point that is exactly halfway between two given points. The formula is \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). By substituting the coordinates, you can find the specific midpoint.
In the given exercise, we use the midpoint formula to calculate the midpoint of the line segment joining \( (8, 3\sqrt{5}) \) and \( (-6, 7\sqrt{5}) \). After substituting and simplifying, the midpoint is \( (1, 5\sqrt{5}) \).
The distance formula, on the other hand, is used to find how far apart two points are. Though not used directly in this problem, it complements the midpoint formula by providing deeper insight into spatial relationships. Both concepts together enhance our understanding of geometric structures.
The **midpoint formula** is specifically used to find the point that is exactly halfway between two given points. The formula is \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). By substituting the coordinates, you can find the specific midpoint.
In the given exercise, we use the midpoint formula to calculate the midpoint of the line segment joining \( (8, 3\sqrt{5}) \) and \( (-6, 7\sqrt{5}) \). After substituting and simplifying, the midpoint is \( (1, 5\sqrt{5}) \).
The distance formula, on the other hand, is used to find how far apart two points are. Though not used directly in this problem, it complements the midpoint formula by providing deeper insight into spatial relationships. Both concepts together enhance our understanding of geometric structures.
Other exercises in this chapter
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