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Chapter 1: Problem 4

Given a rectangle, construct a square with the same content.

Short Answer

Expert verified
Use the side lengths of the rectangle to calculate the side of the square as the square root of the rectangle's area (\(c = \sqrt{a*b}\)). You can then draw the square using this length.

Step by step solution

01

Understand the problem

We're given a rectangle, and from this rectangle, a square with the same area or content needs to be constructed. Let's denote the sides of the rectangle as \(a\) and \(b\), with \(a\) being the length and \(b\) the width. The area of this rectangle would be \(a*b\). The square that we want to construct should also have the same area, meaning the side of the square (let's call it \(c\)) should be such that \(c^2 = a*b\).
02

Calculate the side of the square

To find the side of the square, we need to find a number \(c\) that, when squared, equals to the area of the rectangle. In other words, we want to solve for \(c\) in the equation \(c^2 = a*b\). This translates to \(c = \sqrt{a*b}\).
03

Construct the square

Once the length of the side of the square (\(c\)) has been calculated, we can construct the square. This can be done using a straight edge to draw four lines of equal length, each equal to \(c\), and each meeting at a right angle with its adjacent lines.

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Key Concepts

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

Rectangle
A rectangle is a four-sided shape, often seen in everyday objects like books and screens. It has opposite sides that are equal in length, meaning that two sides are longer (the length) and two sides are shorter (the width). The defining feature is that each of its four angles is a right angle, or 90 degrees.
  • The formula for calculating the area of a rectangle is length times width, written as \(a \times b\), where \(a\) is the length and \(b\) is the width.
  • It's crucial to know these dimensions, as they directly determine how much space the rectangle covers.
Understanding rectangles is often the first step in geometric constructions, setting the foundation for converting them into other shapes, like squares.
Square
A square is a unique type of rectangle, where all four sides are the same length. This makes it both a special rectangle and a four-sided regular polygon. Every angle in a square is a right angle, just like in a rectangle. However, squares have additional symmetry due to their equal sides.
  • This symmetry makes them useful in many geometric problems and constructions, such as tiling patterns and engineering designs.
  • The perimeter of a square can be found by multiplying the side length by four, or \(4 \times c\), where \(c\) is the side length.
When constructing a square from another shape, like a rectangle, it's important to keep all sides equal and ensure each angle is 90 degrees.
Area
Area is a measure of how much space a shape covers, expressed in square units. For rectangles and squares, calculating area helps us understand the space these shapes occupy.
  • For rectangles, the formula is the product of the length and width: \(a \times b\).
  • For squares, the area is found by squaring the side length, \(c^2\).
  • Equating the areas of different shapes allows us to compare and construct one shape into another with the same area.
In the case of creating a square from a rectangle with equal area, you must compute the side of the square by taking the square root of the rectangle's area, converting the space accurately while maintaining the content.

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Most popular questions from this chapter

Let \(A B C D\) be a quadrilateral. Show that the figure formed by joining the midpoints of the four sides is a parallelogram.

Using a ruler and rusty compass, given a segment \(A B\) and given a ray \(A C,\) construct a point \(D\) on the ray \(A C\) such that \(A B \cong A D\) (IMAGE CAN'T COPY).

Given a segment \(A B\), given an angle \(\alpha\), and given another segment \(d\), construct a triangle \(A B C\) with base equal to \(A B\) angle \(\alpha\) at \(C,\) and such that \(A C+B C=d\) (IMAGE CAN'T COPY).

(Painting the plane). If the plane has been colored so that each point has one of three colors (red, yellow, blue), prove that for any interval \(A B\) there exist two points \(C, D\) of the same color, with \(A B \cong C D\). (It is an unsolved problem whether the same result is true for four colors.)

Use cyclic quadrilaterals to give another proof of Proposition \(5.6,\) as follows. Let \(A B C\) be the given triangle. Let the altitudes \(B L\) and \(C K\) meet at \(H .\) Let \(A H\) meet the opposite side at \(M .\) Then show that \(A M \perp B C\). (This proof is probably the one known to Archimedes.) (Figure can't copy)
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