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Chapter 5: Problem 161

Prove that any parallelogram \(A B C D\) has the same area as the rectangle on the same base \(D C\) and "with the same height" (i.e. lying between the same two parallel lines \(A B\) and \(D C)\).

Short Answer

Expert verified
The parallelogram and the rectangle have the same area because they share the same base and height.

Step by step solution

01

Understanding the Problem

We are given a parallelogram \(ABCD\) and need to prove that its area is equal to a rectangle that has the same base \(DC\) and the same height. Both figures lie between the same parallel lines \(AB\) and \(DC\).
02

Recall the Area of Parallelogram

The area \(A_p\) of a parallelogram is calculated using the formula \(A_p = \, base \, \times \, height\). For parallelogram \(ABCD\), the base can be taken as \(DC\) and the height is the perpendicular distance between the lines \(AB\) and \(DC\).
03

Calculate the Area of the Rectangle

Similarly, the area \(A_r\) of a rectangle is given by \(A_r = \, base \, \times \, height\). If we construct a rectangle on base \(DC\) and between the lines \(AB\) and \(DC\), it will have the same base and height as the parallelogram.
04

Comparison of Areas

By comparing the formulas, both the parallelogram and the rectangle have base \(DC\) and the same height (distance between \(AB\) and \(DC\)). Therefore, \(A_p = A_r\). This shows that they have the same area, proving the initial statement.

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Key Concepts

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

Area of Parallelogram
The area of a parallelogram is determined by its base and height. Imagine slicing the parallelogram with a straight line from one edge to the opposite. You'll notice there are two sets of parallel sides.
  • Base: Choose any side to be the base. In the parallelogram ABCD, we can take DC as the base.
  • Height: This is the perpendicular distance between the chosen base DC and its opposite parallel line AB. This isn't always the side length but instead, the shortest distance between the parallel lines.
With the formula, the area is calculated as:\[A_p = \, \text{base} \, \times \, \text{height}\]This means the space covered by the parallelogram is only dependent on these two measures, perfectly ignoring slant or angles of the sides. This beauty makes it easy to relate with other rectangular shapes.
Rectangle Area
Calculated the same way, a rectangle's area uses its length and width. These are simply other names for its base and height:
  • Base: Any side can be considered the base, similar to the parallelogram.
  • Height: The side perpendicular to the chosen base.
For rectangles, the base and height meet at perfect right angles, simplifying calculations:\[A_r = \, \text{base} \, \times \, \text{height}\]Rectangles represent the most straightforward concept of the area as each angle is a right angle, making visualizing and contrasting with other shapes very intuitive. When comparing a rectangle built with the same base and height as a parallelogram, there's a simple observation 鈥� they have the same area.
Parallel Lines
Parallel lines are two lines that run beside each other, equidistant throughout. In geometry, they never meet no matter how long they extend.
  • Definition: Lines that are always the same distance apart and will never converge.
  • Implications in Geometry: Allow for consistent measurements across various shapes, like parallelograms, where their opposite sides are parallel.
In the context of parallelograms, these parallel lines (e.g., AB and DC) form the frame within which shapes can share the same height. This property makes it invaluable for geometric proofs where area calculations are required, as they provide a stable base and height determination for both the parallelogram and the corresponding rectangle.
Geometric Proofs
Geometric proofs are the backbone of mathematical reasoning, using logic and known facts to demonstrate the truth of a statement.
  • Importance: They provide absolute certainty in geometry, which supports conclusions in various applications.
  • Process: Breaks down into simple known principles. Start from axioms, follow through with logical reasoning and reach a conclusion.
For proving the area relationship between a parallelogram and a rectangle, we begin by understanding both areas are defined by base and height. By aligning the parallelogram and a corresponding rectangle between the same parallel lines, the proof becomes evident as their measures coincide. This logical comparison highlights how geometric proofs can simplify complex ideas, building deeper understanding of spatial relationships without invoking algebra or numerical examples.

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Most popular questions from this chapter

You are given a pyramid \(A B C D\) with all three faces meeting at \(A\) being right angled triangles with right angles at \(A .\) Suppose \(\underline{A B}=b,\) \(\underline{A C}=c, \underline{A D}=d\) (a) Calculate the areas of \(\triangle A B C, \triangle A C D, \triangle A D B\) in terms of \(b, c, d\). (b) Calculate the area of \(\triangle B C D\) in terms of \(b, c, d\). (c) Compare your answer in part (b) with the sum of the squares of the three areas you found in part (a).

Suppose that the line \(X A Y\) is tangent to the circumcircle of \(\triangle A B C\) at the point \(A,\) and that \(X\) and \(C\) lie on opposite sides of the line \(A B .\) Prove that \(\angle X A B=\angle A C B\)

(a) Find a formula for the surface area of a right circular cone with base of radius \(r\) and slant height \(l\). (b) Find a similar formula for the surface area of a right pyramid with apex \(A\) whose base \(B C D E \cdots\) is a regular \(n\) -gon with inradius \(r\).

(a) Given points \(A, B,\) with \(\underline{A B}=2 c\) and a real number \(a>c .\) Find the locus of all points \(X\) such that \(\underline{A X}+\underline{B X}=2 a\) (b) Given a point \(F\) and a line \(m,\) find the locus of all points \(X\) such that the ratio distance from \(X\) to the point \(F:\) distance from \(X\) to the line \(m\) is a positive constant \(e<1\) (c) Prove that parts (a) and (b) give different ways of specifying the same curve, or locus.

Let \(\triangle A B C\) be an acute angled triangle. (a) Prove that, among all possible triangles \(\triangle P Q R\) inscribed in \(\triangle A B C,\) with \(P\) on \(B C, Q\) on \(C A, R\) on \(A B,\) the orthic triangle is the one with the shortest perimeter. (b) Suppose that the sides of \(\triangle A B C\) act like mirrors. A ray of light is shone along one side of the orthic triangle \(P Q,\) reflects off \(C A,\) and the reflected beam then reflects in turn off \(A B\). Where does the ray of light next hit the side \(B C ?\) (Alternatively, imagine the sides of the triangle as billiard table cushions, and explain the path followed by a ball which is projected, without spin, along \(P Q .)\)
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