/*! 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 51 Show that a square whose diagona... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 51

Show that a square whose diagonal has length \(d\) has perimeter \(2 \sqrt{2} d\).

Short Answer

Expert verified
Since the square can be divided into two right-angled triangles by its diagonal, we can use Pythagorean theorem to find the side length 's' in terms of diagonal 'd': \(s = \frac{d\sqrt{2}}{2}\). Then, calculate the perimeter 'P' using \(P = 4s\), which gives \(P = 2d\sqrt{2}\). Therefore, a square whose diagonal has length 'd' has a perimeter of \(2 \sqrt{2} d\).

Step by step solution

01

Draw a square and its diagonal

Draw a square with side length 's' and let the diagonal be of length 'd'. Observe that the square can be divided into two right-angled triangles by its diagonal.
02

Use the Pythagorean theorem

In one of the right-angled triangles formed by the diagonal of the square, the diagonal 'd' is the hypotenuse, and both sides of the square 's' act as the perpendicular and base which have equal lengths (since it's a square). Hence, we can write the Pythagorean theorem as: \(s^2 + s^2 = d^2\).
03

Solve for the side length 's'

We have the equation: \(2s^2 = d^2\). Now, to find the side length 's' in terms of the diagonal 'd', we can rearrange the equation as: \(s^2 = \frac{1}{2}d^2\). Taking the square root of both sides, we get: \(s = \sqrt{\frac{1}{2}d^2}\).
04

Simplify the expression for 's'

The expression for 's' can be simplified as: \(s = \sqrt{\frac{d^2}{2}} = \frac{d}{\sqrt{2}} \times \sqrt{\frac{2}{2}} = \frac{d\sqrt{2}}{2}\).
05

Calculate the perimeter of the square

Now that we have the side length 's' in terms of the diagonal 'd', we can calculate the perimeter 'P' of the square using the formula: \(P = 4s\). Substituting the expression for 's', we get: \(P = 4 \times \frac{d\sqrt{2}}{2} = 2d\sqrt{2}\). Thus, a square whose diagonal has length 'd' has a perimeter of \(2 \sqrt{2} d\).

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.

Square
A square is a special type of quadrilateral with four equal sides and four right angles. Each side of a square is the same length, forming a perfect symmetrical shape. This unique property of equal sides makes it easy to use mathematical formulas to find other characteristics like the perimeter and diagonal.

When working with squares, it's important to remember:
  • The interior angles are each 90 degrees.
  • The opposite sides are parallel.
  • The diagonals of a square are equal in length and intersect at right angles, dividing the square into two congruent right-angled triangles.
These properties allow us to apply the Pythagorean theorem conveniently when looking at problems involving squares.
Diagonal
The diagonal of a square is a line segment that connects any two opposite corners and divides the square into two right-angled triangles. The length of the diagonal is a crucial component when using the Pythagorean theorem, which becomes applicable because the square's diagonal acts as the hypotenuse of these right-angled triangles.

To calculate the diagonal of a square with side length 's', we utilize the formula derived from the Pythagorean theorem:
  • The relationship between the side length 's' and the diagonal 'd' is given by: \(s^2 + s^2 = d^2\).
  • Simplified, this becomes \(2s^2 = d^2\), leading to \(d = s\sqrt{2}\).
Understanding how the diagonal relates to the side length helps in solving various geometric problems concerning squares.
Perimeter
The perimeter of a square is the total distance around its four sides. This is a straightforward concept for squares since all sides are equal, allowing us to easily find the perimeter by multiplying the side length by four. However, sometimes, like in our exercise, the problem involves giving the perimeter in terms of the diagonal.

By knowing the relationship between the diagonal and side length of a square (\(s = \frac{d\sqrt{2}}{2}\)), we substitute this expression into the perimeter formula:
  • The perimeter \(P\) is calculated as \(P = 4s\).
  • Substituting for 's' we get: \(P = 4 \times \frac{d\sqrt{2}}{2} = 2d\sqrt{2}\).
This computation elegantly shows how we can express perimeter in terms of the diagonal, showcasing the beautiful symmetries inherent in squares.

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

Estimate the indicated value without using a calculator. $$ \ln 3.0012-\ln 3 $$

How long does it take for money to increase by a factor of five when compounded continuously at \(7 \%\) per year?

Show that for every number \(c\), we have $$ e^{c+t}-e^{c} \approx t e^{c} $$ for small values of \(t\)

Explain why the two previous problems imply that \(\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)\) is the midpoint of the line segment with endpoints \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\)

The functions cosh and \(\sinh\) are defined by $$ \cosh x=\frac{e^{x}+e^{-x}}{2} \text { and } \sinh x=\frac{e^{x}-e^{-x}}{2} $$ for every real number \(x .\) For reasons that do not concern us here, these functions are called the hyperbolic cosine and hyperbolic sine; they are useful in engineering. Show that the range of \(\sinh\) is the set of real numbers.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

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.