/*! 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 94 Show that the points \(A(1,1+d),... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 94

Show that the points \(A(1,1+d), B(3,3+d),\) and \(C(6,6+d)\) are collinear (lie along a straight line) by showing that the distance from \(A\) to \(B\) plus the distance from \(B\) to \(C\) equals the distance from \(A\) to \(C\).

Short Answer

Expert verified
Yes, the given points A, B and C are collinear. The calculation has shown that AB + BC = AC, which means that these points lie along a straight line.

Step by step solution

01

Calculate the Distance from Point A to Point B

Apply the distance formula, which for two points (x1, y1) and (x2, y2), is \(d = \sqrt{(x2-x1)^2 + (y2-y1)^2}\), to find distance from Point A to B. With A(1, 1+d) and B(3, 3+d) we get distance AB = \(\sqrt{(3-1)^2+(3+d-(1+d))^2} = 2\sqrt{2}\)
02

Calculate the Distance from Point B to Point C

Apply the distance formula similarly to find distance from Point B to C. With B(3, 3+d) and C(6, 6+d) we get distance BC = \(\sqrt{(6-3)^2+(6+d-(3+d))^2} = 3\sqrt{2}\)
03

Calculate the Distance from Point A to Point C

Apply once again the distance formula to find distance from Point A to C. With A(1, 1+d) and C(6, 6+d) we get distance AC = \(\sqrt{(6-1)^2+(6+d-(1+d))^2} = 5\sqrt{2}\)
04

Show That AB + BC = AC

Add the distances from A to B and B to C, and show that it is equal to the distance from A to C. Therefore, 2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}
05

Conclusion

Since AB + BC equals AC, we can conclude that the points A, B, and C lie along a straight line, meaning they are collinear as per the definition of collinear points.

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

Here is the 2011 Federal Tax Rate Schedule \(X\) that specifies the tax owed by a single taxpayer. (TABLE CAN'T COPY) The preceding tax table can be modeled by a piecewise function, where \(x\) represents the taxable income of a single taxpayer and \(T(x)\) is the tax owed: $$T(x)=\left\\{\begin{array}{c}0.10 x \\\850.00+0.15(x-8500) \\\4750.00+0.25(x-34,500) \\\17,025.00+0.28(x-83,600) \\\\\frac{?}{?}\end{array}\right.$$ if \(\quad 0 < x \leq 8500\) if \(\quad 8500 < x \leq 34,500\) if \(\quad 34,500 < x \approx 83,600\) if \(\quad 83,600 < x =174,400\) if \(174,400 < x \leq 379,150\) if \(\quad x >379,150\). Use this information to solve. Find and interpret \(T(20,000)\)

How can a graphing utility be used to visually determine if two functions are inverses of each other?

a. Use a graphing utility to graph \(f(x)=x^{2}+1\) b. Graph \(f(x)=x^{2}+1, g(x)=f(2 x), h(x)=f(3 x),\) and \(k(x)=f(4 x)\) in the same viewing rectangle. c. Describe the relationship among the graphs of \(f, g, h\) and \(k,\) with emphasis on different values of \(x\) for points on all four graphs that give the same \(y\) -coordinate. d. Generalize by describing the relationship between the graph of \(f\) and the graph of \(g,\) where \(g(x)=f(c x)\) for \(c>1\) e. Try out your generalization by sketching the graphs of \(f(c x)\) for \(c=1, c=2, c=3,\) and \(c=4\) for a function of your choice.

Use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. In addition, graph the line \(y-x\) and visually determine if \(f\) and g are inverses. $$ f(x)=4 x+4, g(x)=0.25 x-1 $$

What must be done to a function's equation so that its graph is reflected about the \(y\) -axis?
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Statistics

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.