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Chapter 2: Problem 11

For the points \(P\) and \(Q,\) find ( \(a\) ) the distance \(d(P, Q)\) and ( \(b\) ) the coordinates of the midpoint of the segment \(P Q .\) See Examples 2 and \(5(a)\) $$P(-5,-7), Q(-13,1)$$

Short Answer

Expert verified
(a) The distance is \( 8\sqrt{2} \). (b) The midpoint is \( (-9, -3) \).

Step by step solution

01

Understand the problem

We need to find two things: (a) the distance between points \( P \) and \( Q \), and (b) the coordinates of the midpoint of the segment connecting \( P \) and \( Q \).
02

Use the distance formula

The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \[ d(P, Q) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]. Here, \( P(-5,-7) \) and \( Q(-13,1) \).
03

Substitute the coordinates into the distance formula

Plug in the coordinates of \( P \) and \( Q \): \[(x_1, y_1) = (-5,-7) \] and \[(x_2, y_2) = (-13,1) \]. Calculate \[(x_2 - x_1) = (-13) - (-5) = -8 \] and \[(y_2 - y_1) = 1 - (-7) = 8 \]. So, \[ d(P, Q) = \sqrt{(-8)^2 + 8^2} = \sqrt{64 + 64} = \sqrt{128} = 8\sqrt{2} \].
04

Use the midpoint formula

The midpoint \( M \) of a segment connecting points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \[( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}) \].
05

Substitute the coordinates into the midpoint formula

Plug in the coordinates of \( P \) and \( Q \): \[(x_1, y_1) = (-5,-7) \] and \[(x_2, y_2) = (-13,1) \]. Calculate \[( \frac{-5 + (-13)}{2}, \frac{-7 + 1}{2} ) = ( \frac{-18}{2}, \frac{-6}{2} ) = (-9, -3) \]. So, the coordinates of the midpoint are \( (-9, -3) \).

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Key Concepts

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

Distance Between Points
Calculating the distance between two points in a plane is an essential skill in geometry and precalculus. To find the distance, you use the distance formula. The distance formula is derived from the Pythagorean theorem and is given by \( d(P, Q) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
Midpoint of a Segment
The midpoint of a segment is the exact middle point between two endpoints. To find the midpoint, you use the midpoint formula. This formula calculates the average of the x-coordinates and the y-coordinates of the endpoints. The formula is given by \( M(x, y) = \( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \) \). This helps in finding a point that is equidistant from both endpoints.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, connects algebra and geometry through graphs and coordinates. Points are denoted by their coordinates \( (x, y) \). This branch of mathematics allows for calculations of distances, midpoints, slopes, and other geometric properties using algebraic equations.
Precalculus
Precalculus bridges the gap between algebra, geometry, and calculus. It involves the study of functions, complex numbers, trigonometry, and coordinates. Mastering these core concepts is essential for understanding calculus. Both the distance and midpoint formulas are fundamental tools in precalculus, helping solve a variety of problems involving points and segments.

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Most popular questions from this chapter

Write an equation ( \(a\) ) in standard form and ( \(b\) ) in slope-intercept form for the line described. See Example \(6 .\) Find \(k\) so that the line through \((4,-1)\) and \((k, 2)\) is (a) parallel to \(3 y+2 x=6\) (b) perpendicular to \(2 y-5 x=1\)

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