/*! 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 89 Proof Prove that the figure form... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 89

Proof Prove that the figure formed by connecting consecutive midpoints of the sides of any quadrilateral is a parallelogram.

Short Answer

Expert verified
The figure formed by connecting consecutive midpoints of the sides of any quadrilateral is a parallelogram because the opposite sides are parallel to each other.

Step by step solution

01

Define the Quadrilateral

Label the four vertices of the quadrilateral \( A, B, C,\) and \( D\). Label the midpoints of each side as follows: midpoint of \( AB\) as \( E\), midpoint of \( BC\) as \( F\), midpoint of \( CD\) as \( G\), and midpoint of \( DA\) as \( H\). Now, by connecting the midpoints \( E, F, G,\) and \( H\), a new figure is formed inside the quadrilateral.
02

Prove that Opposite Sides are Parallel

By the midpoint theorem, a line drawn through the midpoints of two sides of a triangle is parallel to the third side and equal to half of it. Therefore, \( EF\) is parallel to \( GH\) and equal to half of \( AC\). Similarly, \( HG\) is parallel to \( EF\) and equal to half of \( BD\). Therefore, \( EF\) is parallel to \( GH\), and \( EH\) is parallel to \( FG\).
03

Conclusion

Since there is a pair of parallel sides on both accounts, the figure \(EFGH\) is a parallelogram.

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

Sketching a Graph of a Function In Exercises \(33-40\) , sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. $$ g(x)=\frac{4}{x} $$

Sketching a Graph In Exercises \(87-90,\) sketch a possible graph of the situation. The speed of an airplane as a function of time during a 5 -hour flight

Deciding Whether an Equation Is a Function In Exercises \(47-50\) , determine whether \(y\) is a function of \(x .\) $$ x^{2}+y^{2}=16 $$

Finding the Domain and Range of a Function In Exercises \(11-22,\) find the domain and range of the function. $$ f(x)=\sqrt{16-x^{2}} $$

Let \(R\) be the region consisting of the points \((x, y)\) of the Cartesian plane satisfying both \(|x|-|y| \leq 1\) and \(|y| \leq 1 .\) Sketch the region \(R\) and find its area.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Mechanics 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.