/*! 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 6 Show that the midpoint of the li... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 6

Show that the midpoint of the line segment joining the points \((a, b)\) and \((c, d)\) is \(((a+c) / 2,(b+d) / 2)\).

Short Answer

Expert verified
Using the midpoint formula M = \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\), and substituting the given points \((a, b)\) and \((c, d)\), we find that the midpoint M = \(\left(\frac{a + c}{2}, \frac{b + d}{2}\right)\), thus proving that the midpoint of the line segment joining \((a, b)\) and \((c, d)\) is indeed \(((a+c) / 2, (b+d) / 2)\).

Step by step solution

01

Recall the midpoint formula

The midpoint formula is used to find the coordinates of the midpoint of a line segment given the coordinates of its two endpoints. It is derived from the concept of averages and is given by the following formula: For two points \((x_1, y_1)\) and \((x_2, y_2)\), the midpoint M has the coordinates \((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\).
02

Substitute the given points into the midpoint formula

Now, substitute the given points \((a, b)\) and \((c, d)\) into the midpoint formula: M = \(\left(\frac{a + c}{2}, \frac{b + d}{2}\right)\).
03

Conclusion

Thus, we have shown that the midpoint of the line segment joining the points \((a, b)\) and \((c, d)\) is indeed \(((a+c) / 2, (b+d) / 2)\).

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!

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

The vectors \(u_{1}=(1,1,1,1), u_{2}=(0,1,1,1), u_{3}=(0,0,1,1)\), and \(u_{4}=(0,0,0,1)\) form a basis for \(F^{4}\). Find the unique representation of an arbitrary vector \(\left(a_{1}, a_{2}, a_{3}, a_{4}\right)\) in \(\mathrm{F}^{4}\) as a linear combination of \(u_{1}, u_{2}, u_{3}\), and \(u_{4}\).

Show that if \(S_{1}\) and \(S_{2}\) are arbitrary subsets of a vector space V, then $\operatorname{span}\left(S_{1} \cup S_{2}\right)=\operatorname{span}\left(S_{1}\right)+\operatorname{span}\left(S_{2}\right)$. (The sum of two subsets is defined in the exercises of Section 1.3.)

Show that if \(S_{1}\) and \(S_{2}\) are subsets of a vector space \(\mathrm{V}\) such that \(S_{1} \subseteq S_{2}\), then $\operatorname{span}\left(S_{1}\right) \subseteq \operatorname{span}\left(S_{2}\right) .\( In particular, if \)S_{1} \subseteq S_{2}\( and \)\operatorname{span}\left(S_{1}\right)=\mathrm{V}$, deduce that span \(\left(S_{2}\right)=\mathrm{V}\). Visit goo.gl/Fi8Epr for a solution.

Let \(v_{1}, v_{2}, \ldots, v_{k}, v\) be vectors in a vector space \(\mathrm{V}\), and define \(\mathrm{W}_{1}=\) $\operatorname{span}\left(\left\\{v_{1}, v_{2}, \ldots, v_{k}\right\\}\right)$, and $\mathrm{W}_{2}=\operatorname{span}\left(\left\\{v_{1}, v_{2}, \ldots, v_{k}, v\right\\}\right)$. (a) Find necessary and sufficient conditions on \(v\) such that \(\operatorname{dim}\left(\mathrm{W}_{1}\right)=\) \(\operatorname{dim}\left(\mathrm{W}_{2}\right)\). (b) State and prove a relationship involving \(\operatorname{dim}\left(\mathrm{W}_{1}\right)\) and \(\operatorname{dim}\left(\mathrm{W}_{2}\right)\) in the case that $\operatorname{dim}\left(\mathrm{W}_{1}\right) \neq \operatorname{dim}\left(\mathrm{W}_{2}\right)$.

Write \(v=(2,5)\) as a linear combination of \(u_{1}\) and \(u_{2},\) where: (a) \(u_{1}=(1,2)\) and \(u_{2}=(3,5)\) (b) \(u_{1}=(3,-4)\) and \(u_{2}=(2,-3)\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Decision Maths

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.