/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 14 \(7-14\) A pair of points is gra... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 14

\(7-14\) A pair of points is graphed. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the midpoint of the segment that joins them. $$ (5,0),(0,6) $$

Short Answer

Expert verified
Distance: \( \sqrt{61} \), Midpoint: \( \left( \frac{5}{2}, 3 \right) \).

Step by step solution

01

Plot the Points

To plot the points (5, 0) and (0, 6) on a coordinate plane, locate the first point (5, 0) by moving 5 units along the x-axis and 0 units along the y-axis. For the second point (0, 6), move 0 units along the x-axis and 6 units up along the y-axis.
02

Use the Distance Formula

The distance formula is used to find the distance between two points, which is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Substituting the given points (5, 0) and (0, 6) into the formula, we have \( d = \sqrt{(0 - 5)^2 + (6 - 0)^2} = \sqrt{(-5)^2 + (6)^2} = \sqrt{25 + 36} = \sqrt{61} \).
03

Find the Midpoint

The midpoint of a segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). For the points (5, 0) and (0, 6), the midpoint is \( M = \left( \frac{5 + 0}{2}, \frac{0 + 6}{2} \right) = \left( \frac{5}{2}, 3 \right) \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Distance Formula
The distance formula in coordinate geometry is an essential tool for finding the distance between two points on a coordinate plane. Imagine two points, like the ends of a line, and we want to know how far apart they are. This is where the distance formula comes into play. To use the distance formula, you need the coordinates of the two points you wish to measure. Let's say you have two points \( (x_1, y_1) \) and \( (x_2, y_2) \).
  • Step 1: Subtract the x-coordinates: \( x_2 - x_1 \)
  • Step 2: Subtract the y-coordinates: \( y_2 - y_1 \)
  • Step 3: Square both results: \( (x_2 - x_1)^2 \) and \( (y_2 - y_1)^2 \)
  • Step 4: Add these squares: \( (x_2 - x_1)^2 + (y_2 - y_1)^2 \)
  • Step 5: Finally, take the square root: \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
In our example with points \( (5, 0) \) and \( (0, 6) \), you follow these steps and find that the distance between them is \( \sqrt{61} \). By practicing these steps, you'll quickly become proficient at calculating distances on a coordinate plane.
Midpoint Formula
The midpoint formula is used when you want to find a point exactly halfway between two other points. It's like finding the center point of a segment. The formula is easy to remember and apply. Given points \( (x_1, y_1) \) and \( (x_2, y_2) \), the midpoint \( M \) has coordinates:
  • For the x-coordinate: \( \frac{x_1 + x_2}{2} \) - Take the average of the x-values.
  • For the y-coordinate: \( \frac{y_1 + y_2}{2} \) - Take the average of the y-values.
By averaging the x-coordinates and the y-coordinates of the given points, you find a new point that sits right in the middle. For our points \( (5, 0) \) and \( (0, 6) \), following the formula gives a midpoint of \( (\frac{5}{2}, 3) \). This midpoint represents a balanced location along the segment connecting the two original points.
Plotting Points
Plotting points is one of the basic yet critical skills in coordinate geometry. It involves placing points on a graph based on their coordinates. Each point is defined by a pair \( (x, y) \), where \( x \) is the horizontal position and \( y \) is the vertical position. Follow these simple steps to plot any given point:
  • Start from the origin \( (0,0) \), the central point where the x-axis and y-axis meet.
  • Move horizontally along the x-axis by the value of the x-coordinate. If the x-coordinate is positive, move to the right; if negative, move to the left.
  • Next, move vertically up or down based on the y-coordinate. If the y-coordinate is positive, move up; if negative, move down.
By following these steps, you can accurately place any point on a grid. In our specific exercise, points \( (5, 0) \) and \( (0, 6) \) are plotted by moving 5 units right from the origin for the first point, and 6 units up for the second point. Mastering the art of plotting will help you visualize problems and find solutions with ease.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Spread of a Disease The rate \(r\) at which a disease spreads in a population of size \(P\) is jointly proportional to the number \(x\) of infected people and the number \(P-x\) who are not infected. An infection erupts in a small town with population \(P=5000\) . (a) Write an equation that expresses r as a function of x. (b) Compare the rate of spread of this infection when 10 people are infected to the rate of spread when 1000 people are infected. Which rate is larger? By what factor? (c) Calculate the rate of spread when the entire population is infected. Why does this answer make intuitive sense?

Some scientists believe that the average surface temperature of the world has been rising steadily. The average surface temperature is given by $$T=0.02 t+8.50$$ where \(T\) is temperature in \(^{\circ} \mathrm{C}\) and \(t\) is years since 1900 . (a) What do the slope and \(T\) -intercept represent? (b) Use the equation to predict the average global surface temperature in 2100 .

Radiation Energy The total radiation energy E emitted by a heated surface per unit area varies as the fourth power of its absolute temperature T. The temperature is 6000 K at the surface of the sun and 300 K at the surface of the earth. (a) How many times more radiation energy per unit area is produced by the sun than by the earth? (b) The radius of the earth is 3960 mi and the radius of the sun is 435,000 mi. How many times more total radiation does the sun emit than the earth?

Find the slope and \(y\)-intercept of the line and draw its graph. \(2 x-5 y=0\)

(a) Find an equation for the line tangent to the circle \(x^{2}+y^{2}=25\) at the point \((3,-4)\) . (See the figure.) (b) At what other point on the circle will a tangent line be parallel to the tangent line in part (a)?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.