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Chapter 0: Problem 28

Find the perimeter and area of the given figure. A square with diagonal 15 meters.

Short Answer

Expert verified
The perimeter is approximately 42.44 meters, and the area is approximately 112.59 square meters.

Step by step solution

01

- Use the Pythagorean Theorem

A square's diagonals split it into two right-angled triangles. Let the side length of the square be denoted as 's'. According to the Pythagorean Theorem, the relationship between the side length and the diagonal 'd' is: \[ d = s\root{2} \ 15 = s\root{2} \ s = \frac{15}{\root{2}} \ s = \frac{15}{1.414} \ s \approx 10.61 \text{ meters} \]
02

- Calculate the Perimeter

The perimeter of a square is given by the formula: \[ P = 4s \ P = 4 \times 10.61 \ P \approx 42.44 \text{ meters} \]
03

- Calculate the Area

The area of a square is given by the formula: \[ A = s^2 \ A \approx 10.61^2 \ A \approx 112.59 \text{ square meters} \]

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Key Concepts

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

Pythagorean Theorem
The Pythagorean Theorem is essential in geometry. It helps us find the relationship between the sides of a right-angled triangle. In a right-angled triangle, the square of the length of the hypotenuse (longest side) is equal to the sum of the squares of the lengths of the other two sides. This can be written as follows:

of squares and triangles. Understanding it helps solve many geometric problems more quickly.
right-angled triangle
A right-angled triangle has one angle exactly 90 degrees. This unique property makes it special. The sides of a right-angled triangle are referred to as the hypotenuse (longest side), and the two legs (shorter sides). The Pythagorean Theorem is always applicable in such triangles.

In the square problem, when we draw a diagonal, the square is divided into two right-angled triangles. This diagonal becomes the hypotenuse, and the sides of the square are the legs. Knowing this simplifies finding the side lengths of the square using the Pythagorean Theorem.
square properties
A square has four equal sides and four right angles (90 degrees each). The diagonals of a square have unique properties:

  • They are equal in length.
  • They bisect each other at right angles (90 degrees).
  • They split the square into two congruent right-angled triangles.

To find the perimeter of a square, multiply one side by four: 10.61 meters>

These properties help in solving different aspects related to squares in geometry.

The area of the square is given by the formula: side length squared = 10.61^2< Understanding these key properties will aid in solving many geometrical exercises involving squares.

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Most popular questions from this chapter

In Exercises \(21-26,\) round your answer to the nearest tenth where necessary. A scale model of an airplane uses a scale of \(\frac{1}{40} .\) If the model exhibits a wingspan of 8 in., what is the actual wingspan of the plane?

Find the perimeter and area of a square with side \(8 \mathrm{ft}\).

Use the formulas given in this section to compute the total surface area and the volume of the figure described. All answers should be rounded to the nearest tenth. A closed right circular cylinder of height \(5 \mathrm{cm}\) and radius \(4 \mathrm{cm}\)

Find the perimeter and area of a semicircle with diameter 10 in.

In Exercises \(21-26,\) round your answer to the nearest tenth where necessary. The corresponding sides of two similar triangles are in the ratio of 4 to \(7 .\) If a side of the smaller triangle is \(5.8 \mathrm{cm},\) find the length of the corresponding side of the larger triangle.
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