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Chapter 0: Problem 26

Find the distance from (3,-6) to the midpoint of the line segment from (1,2) to (7,8)

Short Answer

Expert verified
The distance is approximately 11.05 units.

Step by step solution

01

Determine the midpoint of the line segment

The midpoint of a segment connecting two points \(x_1, y_1\) and \(x_2, y_2\) is given by the formula \( ext{Midpoint} = rac{(x_1 + x_2)}{2}, rac{(y_1 + y_2)}{2} \). Here, the endpoints of the segment are (1,2) and (7,8). Calculate the midpoint:\[ \text{Midpoint} = \left( \frac{1 + 7}{2}, \frac{2 + 8}{2} \right) = \left( \frac{8}{2}, \frac{10}{2} \right) = (4, 5) \]
02

Calculate the distance to the midpoint

Now, find the distance from point (3,-6) to the midpoint (4,5). Use the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) with \( (x_1,y_1) = (3,-6) \) and \( (x_2,y_2) = (4,5) \).\[ d = \sqrt{(4 - 3)^2 + (5 - (-6))^2} = \sqrt{1 + 11^2} = \sqrt{1 + 121} = \sqrt{122} \] Decimal approximation:\[ d \approx 11.05 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Midpoint Formula
The midpoint formula is a fundamental concept in coordinate geometry, commonly used to find the exact middle point between two given points in a plane. Here's how it works:
  • The formula is derived by averaging the x-coordinates and the y-coordinates of the endpoints.
  • If the points have coordinates \( (x_1, y_1) \) and \( (x_2, y_2) \), then the midpoint is computed as \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).
Understanding this formula helps in determining the center point easily without manual plotting.
In the given solution, this concept is vital to locate the midpoint of a segment joining \( (1,2) \) and \( (7,8) \), providing \( (4,5) \) as the central point.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, is a powerful tool for connecting algebra and geometry. It uses a coordinate system to explore geometric properties and relations.
  • In coordinate geometry, points are plotted on a plane using pairs of numbers (coordinates).
  • The Cartesian plane, divided by x and y-axes, is often utilized for visualizing these points.
Every point is represented in the form \( (x, y) \) which denotes its position relative to the origin of the plane.
This allows for the application of algebraic methods to solve geometric problems. The problem in the exercise illustrates coordinate geometry by using coordinates to find both the midpoint of a line segment and the distance between points.
Analytic Geometry
Analytic geometry combines algebraic and geometric principles to analyze locational relationships of shapes in space. This approach enables the solving of geometric problems with algebraic precision.
  • It provides the basis for concepts like distance and slope, crucial for analyzing geometric figures.
  • The distance formula is extensively applied to find the length between two points, derived from the Pythagorean theorem.
Given any two points, the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) can precisely calculate the length between them.
In the exercise example, the distance from point \( (3,-6) \) to \( (4,5) \) was calculated using this formula, providing a result of roughly \( 11.05 \). This illustrates how analytic geometry aids in solving practical problems involving distances.

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