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Chapter 1: Problem 32

For each pair of points find the distance between them and the midpoint of the line segment joining them. $$(0,0),(\pi / 2,1)$$

Short Answer

Expert verified
Distance: \( \frac{\sqrt{\pi^2 + 4}}{2} \), Midpoint: \( \left( \pi / 4, 1 / 2 \right)\).

Step by step solution

01

Identify Point Coordinates

Extract the coordinates of the given points. The first point is \((0,0)\) and the second point is \((\pi / 2, 1)\).
02

Use Distance Formula

To find the distance between the points, use the distance formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]. Substitute the coordinates into the formula: \[d = \sqrt{(\pi / 2 - 0)^2 + (1-0)^2} = \sqrt{(\pi / 2)^2 + 1^2} = \sqrt{(\pi^2 / 4) + 1} = \sqrt{\pi^2 / 4 + 4 / 4} = \sqrt{(\pi^2 + 4) / 4} = \frac{\sqrt{\pi^2 + 4}}{2} \].
03

Use Midpoint Formula

To find the midpoint of the line segment joining the points, use the midpoint formula: \((x_m, y_m) = \left((x_1 + x_2) / 2, (y_1 + y_2) / 2 \right) \). Substitute the coordinates into the formula: \[ (x_m, y_m) = \left( (0 + \pi / 2) / 2, (0 + 1) / 2 \right) = \left( \pi / 4, 1 / 2 \right) \].

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Key Concepts

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

distance formula
The distance formula is a key tool in coordinate geometry. It helps us determine how far apart two points are on a plane. The formula is derived from the Pythagorean theorem and is given by: }
midpoint formula
Finding the midpoint of two points is very useful in many mathematical applications. It tells us the exact middle point on a line segment between these two points. The midpoint formula is easy to use and follows this calculation: We'll place the x and y coordinates into this formula to get the midpoint. This allows us to understand the symmetrical center between two given points. For example, the midpoint between and equal parts. This is very helpful in geometry, where midpoints play important roles in constructions and proofs.
coordinate geometry
Coordinate geometry, also known as analytic geometry, uses the coordinate plane to describe geometric shapes and their properties. Using equations, we can solve various problems involving points, lines, and shapes. A coordinate plane has two perpendicular lines: the x-axis and the y-axis, forming a grid where any point can be identified with an ordered pair placing it on the grid. Coordinate geometry combines algebra and geometry, providing powerful tools to analyze and solve complex geometric problems. For instance, by using the distance and midpoint formulas, as shown in our example, we can easily find distances and midpoints between points on the plane.
distance between points
Calculating the distance between two points in a coordinate plane involves understanding the position of each point and applying the distance formula. This measurement helps in various fields like physics for determining distances in space, engineering for designing structures, or everyday tasks like finding how far one needs to travel. By locating the two points on a coordinate plane, you measure the straight-line distance, which is the shortest path between them. For example, when we found the distance between the distance formula to get the answer. This provides a concrete way to quantify the space between points, something invaluable in both theoretical and practical applications.

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