Problem 620 Inscribe a square in a given sem... [FREE SOLUTION]

Chapter 34: Problem 620

Inscribe a square in a given semicircle.

Short Answer

Expert verified

To inscribe a square in a given semicircle, first draw the semicircle and diameter AB with center O. Mark the midpoint C of diameter AB, construct two perpendicular bisectors to AB at points A and C, meeting at point P. Then, construct a perpendicular bisector to segment PB at point B, intersecting the semicircle at point Q. The vertices of the inscribed square are A, P, B, and Q, with side length s, where \(2s = d\) (d is the diameter of the semicircle).

Step by step solution

Draw the semicircle and diameter

Draw a semicircle with center O and diameter AB. The diameter AB will act as the base of the inscribed square and will further help in positioning the vertices of the square.

Mark the midpoint of the diameter

Mark the midpoint C of diameter AB. Since the square is inscribed in the semicircle, points A and B will be the opposite vertices of the square, and its side length will be half the diameter. So, AC = CB = s, where s is the side length of the square.

Construct perpendicular bisectors to diameter AB

Construct two perpendicular bisectors to diameter AB at points A and C. They will intersect at point P, which will be the third vertex of the inscribed square. Angle ACB will now be 90 degrees since A, C, and B are part of the inscribed square.

Find the fourth vertex by constructing a perpendicular bisector

Construct a perpendicular bisector to segment PB at point B, which will intersect the semicircle at point Q, which will be the fourth vertex of the inscribed square.

Determine the side length of the square

Since we now have all four vertices of the inscribed square (A, P, B, Q), we can determine the side length of the square by finding the length of segment AC, CB, or any other side. Each side length will be half the diameter of the semicircle. The inscribed square in the given semicircle is now successfully constructed with vertices A, P, B, and Q. The relationship between the side length of the square and the diameter of the semicircle is 2s = d, where s is the side length of the square and d is the diameter of the semicircle.

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Key Concepts

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

Inscribed Shapes

In geometry, an inscribed shape refers to a figure placed within another shape such that all its vertices touch the boundary of the surrounding shape. In this exercise, a square is inscribed in a semicircle. Inscribed shapes are used to explore and solve different problems relating to symmetry, space utilization, and efficiency in geometric configurations. They are often employed in architectural designs and puzzles.

Key points to remember about inscribed shapes are:

Each vertex of the inscribed shape will lie on the boundary of the enclosing shape.
The inscribed shape will be fully contained within the bounding shape without overlapping beyond its edges.

This concept helps us understand relationships between different geometrical figures and aids in visualizing spatial arrangements.

Semicircles

A semicircle is half of a circle, formed by dividing a circle into two equal parts along its diameter. In this problem, a semicircle is the boundary within which a square is inscribed. Understanding semicircles is crucial for this exercise because its diameter serves as a basis for constructing the square.

Important characteristics of semicircles include:

The diameter is a line segment passing through the center that divides the circle into two equal halves.
The center of the semicircle is the midpoint of its diameter.

These characteristics guide the construction of the inscribed square. By knowing that the diameter is part of the semicircle's boundary, we can derive relationships between its length and the inscribed square's dimensions.

Squares

A square is a four-sided polygon, also known as a quadrilateral, with all sides of equal length and all angles being right angles (90 degrees). In this context, the square is inscribed in a semicircle. Knowing the properties of squares helps us place them accurately within the semicircle.

Here are the main features of squares:

All sides are equal in length, which simplifies calculations when inscribing them in another shape.
All angles are equal to 90 degrees, forming right angles at each corner.

The inscribed square relates directly to the diameter of the semicircle, where the diameter touches the square's diagonal. Understanding how to set the square inside the semicircle aligns with recognizing these fundamental properties.

Perpendicular Bisectors

Perpendicular bisectors play a vital role in constructing the inscribed square in this exercise. A perpendicular bisector divides a line segment into two equal parts at a right angle (90 degrees). To inscribe the square within the semicircle, perpendicular bisectors are built to locate the vertices of the square.

Key aspects of perpendicular bisectors include:

They are used to accurately find points equidistant from endpoints of a line segment or diameter in this case.
The intersection points of perpendicular bisectors can help locate vertices of other geometric shapes.

By constructing the perpendicular bisectors of segments formed from the semicircle's diameter, we can determine the exact position of the square's vertices, ensuring that the square fits perfectly within the semicircle.

91影视

Inscribe a square in a given semicircle.

Short Answer

Step by step solution

Draw the semicircle and diameter

Mark the midpoint of the diameter

Construct perpendicular bisectors to diameter AB

Find the fourth vertex by constructing a perpendicular bisector

Determine the side length of the square

Key Concepts

Inscribed Shapes

Semicircles

Squares

Perpendicular Bisectors

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