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Chapter 1: Problem 115

Find the exact length of the diagonal of a square whose sides are 5 feet each.

Short Answer

Expert verified
The exact length of the diagonal is 5\sqrt{2} feet.

Step by step solution

01

Understand the Diagonal Formula

The formula to find the diagonal of a square is given by the equation \(d = s\sqrt{2}\), where \(d\) is the diagonal length and \(s\) is the length of a side of the square.
02

Substitute the Side Length

Given that each side of the square is 5 feet, substitute \(s = 5\) into the diagonal formula: \(d = 5\sqrt{2}\).
03

Simplify the Expression

Simplify the expression to find the exact length of the diagonal: \[d = 5\sqrt{2}\ \text{feet}\].

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Key Concepts

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

Exact Length
To find the exact length of the diagonal of a square, we avoid rounding or approximations. The goal is to use exact values in our mathematical expressions. For instance, when we compute the diagonal of a square with 5-foot sides, we use \(5\sqrt{2}\) instead of using decimal representations.
This approach ensures precision and accuracy in our results. By sticking to the exact values, we communicate the most precise measurement possible.
Exact lengths are particularly useful in fields requiring high precision like architecture and engineering.
Pythagorean Theorem
The Pythagorean Theorem is pivotal in calculating the diagonal of a square. This theorem states that in a right-angled triangle, the square of the hypotenuse is the sum of the squares of the other two sides: \[ c^2 = a^2 + b^2 \]
In the context of a square, each side forms a right-angled triangle with the diagonal as the hypotenuse. Therefore, if each side length (\(s\)) of the square is 5 feet, then: \[ d^2 = 5^2 + 5^2 \]
Simplifying this, we get: \[ d^2 = 25 + 25 = 50 \] Thus, the length of the diagonal \(d\) is \[ d = \sqrt{50} = 5\sqrt{2} \] This makes use of the Pythagorean theorem to find the diagonal's exact length.
Square Geometry
A square has several unique properties which make calculating the diagonal straightforward. Each square is a special type of rectangle where all four sides are of equal length.
Key aspects of square geometry include:
  • All four sides are equal in length.
  • All four interior angles are right angles (90 degrees).
  • The diagonals bisect each other at right angles and are equal in length.
In terms of diagonal calculations, knowing these properties allows us to accurately apply the Pythagorean theorem. Given a square with side length \(s\), the diagonal divides the square into two equal right-angled triangles, each with legs \(s\) and hypotenuse \(s\sqrt{2}\).
Diagonal Formula
The formula to find the diagonal of a square stems directly from the Pythagorean theorem. Knowing the side length \(s\), the diagonal \(d\) is calculated as: \[ d = s\sqrt{2} \]
This formula simplifies diagonal length calculations quickly and effectively. For example, in a square with each side 5 feet, substitute \(s = 5\): \[ d = 5\sqrt{2} \]
This yields the exact length of the diagonal, avoiding the need for approximations.
By using \(s\sqrt{2}\), we make the computation both simple and exact.

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