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Chapter 1: Problem 15

The corners of a square lie on a circle of diameter \(D=0.35 \mathrm{m}\) Each side of the square has a length \(L\). Find \(L\).

Short Answer

Expert verified
The side length \(L\) is approximately 0.2475 meters.

Step by step solution

01

Understanding the Problem

We have a circle with a given diameter, and we need to find the side length of a square inscribed in this circle. The diameter of the circle is given as 0.35 meters. The side length of the square is denoted by \(L\).
02

Relating Diameter and Diagonal

When a square is inscribed in a circle, the diagonal of the square is equal to the diameter of the circle. Therefore, the diagonal of our square is also 0.35 meters.
03

Using the Diagonal to Find the Side Length

For a square, if the diagonal is \(d\) and the side length is \(L\), the relationship is given by the Pythagorean theorem as \(d = L\sqrt{2}\). Here, \(d = 0.35\) meters.
04

Solving for L

From the equation \(d = L\sqrt{2} = 0.35\), we can solve for \(L\) by dividing the diagonal by \(\sqrt{2}\): \[ L = \frac{0.35}{\sqrt{2}} \] Calculate this to get \(L\).
05

Calculation

Now, calculating the value, \[ L = \frac{0.35}{\sqrt{2}} \approx \frac{0.35}{1.414} \approx 0.2475 \mathrm{m} \] Therefore, the side length \(L\) is approximately 0.2475 meters.

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Key Concepts

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

Inscribed Square
An inscribed square is a square whose corners touch the circumference of a circle. This unique position positions the square perfectly inside the circle. The circle, in this case, serves as the circumscribed circle because it wraps around the square, touching it only at the vertices. The diagonal of the inscribed square becomes a crucial line, as it equals the diameter of the circle encircling the square. This hint is the key to solving for unknown values such as side lengths of the square when the circle's diameter is known.
Circle Properties
Circles have a lot of interesting properties that can help in geometric problems. One such property relevant to inscribed shapes is that any diagonal from a shape inscribed in a circle is equal to the diameter of the circle. This is particularly useful when dealing with inscribed squares or rectangles.
  • A circle's diameter is twice its radius.
  • The circumference of a circle is given by the formula \(C = \pi D\).
  • All points on a circle are equidistant from the center.
These circle properties are fundamental in connecting the circle's dimensions with other shapes, as seen with an inscribed square.
Diagonal of a Square
The diagonal of a square is not only a line connecting opposite corners but also a line that slices the square into two congruent right-angled triangles. The diagonal length can be found using the Pythagorean theorem. It is also crucial when squares are inscribed in circles because the diagonal aligns perfectly with the circle's diameter.
  • The diagonal divides the square into two equal right triangles.
  • The formula for the diagonal \(d\) when the side length \(L\) is given is \(d = L\sqrt{2}\).
  • In the context of a circle, this diagonal becomes equal to the circle's diameter.
This knowledge bridges the gap between the known dimensions of a circle and the required dimensions of an inscribed square.
Pythagorean Theorem
The Pythagorean theorem is a cornerstone of geometry, especially useful when dealing with right triangles. It relates the three sides of a right triangle, providing a way to calculate one side when the other two are known.
  • For a triangle with sides \(a\), \(b\), and hypotenuse \(c\), the theorem states: \(a^2 + b^2 = c^2\).
  • In a square, which is made up of two congruent right triangles, the diagonal acts as the hypotenuse connecting the two right angles.
  • Applying the theorem, the diagonal \(d\) of a square with side length \(L\) is found by \(d = L\sqrt{2}\).
This fundamental theorem not only helps in finding side lengths of squares using diagonals but also in making connections with other geometric dimensions in inscribed shapes.

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Most popular questions from this chapter

Two bicyclists, starting at the same place, are riding toward the same campground by two different routes. One cyclist rides \(1080 \mathrm{m}\) due east and then turns due north and travels another \(1430 \mathrm{m}\) before reaching the campground. The second cyclist starts out by heading due north for \(1950 \mathrm{m}\) and then turns and heads directly toward the campground. (a) At the turning point, how far is the second cyclist from the campground? (b) In what direction (measured relative to due east) must the second cyclist head during the last part of the trip?

You are driving into St. Louis, Missouri, and in the distance you see the famous Gateway to the West arch. This monument rises to a height of \(192 \mathrm{m}\). You estimate your line of sight with the top of the arch to be \(2.0^{\circ}\) above the horizontal. Approximately how far (in kilometers) are you from the base of the arch?

Displacement vector \(\overrightarrow{\mathbf{A}}\) points due east and has a magnitude of \(2.00 \mathrm{km} .\) Displacement vector \(\overrightarrow{\mathbf{B}}\) points due north and has a magnitude of 3.75 km. Displacement vector \(\overrightarrow{\mathbf{C}}\) points due west and has a magnitude of \(2.50 \mathrm{km} .\) Displacement vector \(\overrightarrow{\mathbf{D}}\) points due south and has a magnitude of \(3.00 \mathrm{km} .\) Find the magnitude and direction (relative to due west) of the resultant vector \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}+\overrightarrow{\mathbf{D}}\).

You and your team are exploring a river in South America when you come to the bottom of a tall waterfall. You estimate the cliff over which the water flows to be about 100 feet tall. You have to choose between climbing the cliff or backtracking and taking another route, but climbing the cliff would cut two hours off of your trip. There is only one experienced climber in the group: she would climb the cliff alone and drop a rope over the edge to lift supplies and allow the others to climb without packs. The climber estimates it will take her 45 minutes to get to the top. However, you are concerned that the rope might be too short to reach the bottom of the cliff (it is exactly \(30.0 \mathrm{m}\) long \() .\) If it is too short, she'll have to climb back down (another 45 minutes) and you will be too far behind schedule to get to your destination before dark. As you contemplate how to determine whether the rope is long enough, you notice that the late afternoon shadow of the cliff grows as the sun descends over its edge. You suddenly remember your trigonometry. You measure the length of the shadow from the base of the cliff to the shadow's edge (\(144 \mathrm{ft}\)), and the angle subtended between the base and top of the cliff measured from the shadow's edge. The angle is \(38.1^{\circ} .\) Do you send the climber, or start backtracking to take another route?

The route followed by a hiker consists of three displacement vectors \(\overrightarrow{\mathbf{A}}, \overrightarrow{\mathbf{B}},\) and \(\overrightarrow{\mathbf{C}}\). Vector \(\overrightarrow{\mathbf{A}}\) is along a measured trail and is \(1550 \mathrm{m}\) in a direction \(25.0^{\circ}\) north of east. Vector \(\overrightarrow{\mathbf{B}}\) is not along a measured trail, but the hiker uses a compass and knows that the direction is \(41.0^{\circ}\) east of south. Similarly, the direction of vector \(\overrightarrow{\mathbf{C}}\) is \(35.0^{\circ}\) north of west. The hiker ends up back where she started. Therefore, it follows that the resultant displacement is zero, or \(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{C}}=\mathbf{0} .\) Find the magnitudes of (a) vector \(\overrightarrow{\mathbf{B}}\) and (b) vector \(\overrightarrow{\mathbf{C}}\).
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