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Chapter 5: Problem 170

Let \(A B C D\) be a trapezium with \(A B \| D C,\) in which \(A B\) has length \(a\) and \(D C\) has length \(b\). (a) Let \(M\) be the midpoint of \(\underline{A D}\) and let \(N\) be the midpoint of \(\underline{B C}\). Prove that \(M N \| A B\) and find the length of \(\underline{M N}\). (b) If the perpendicular distance between \(A B\) and \(D C\) is \(d,\) find the area of the trapezium \(A B C D\).

Short Answer

Expert verified
(a) \(MN = \frac{a+b}{2}\) and \(MN \parallel AB\). (b) Area = \(\frac{1}{2}(a+b)d\).

Step by step solution

01

Establish Midpoint Coordinates

Let the coordinates of point \(A\) be \((0,0)\) and point \(B\) be \((a,0)\). Let the coordinates of \(D\) be \((0, h)\) and \(C\) be \((b, h)\), because \(AB \parallel DC\), they must lie on horizontal lines. The midpoints \(M\) and \(N\) are therefore \(M = \left(\frac{0+0}{2}, \frac{0+h}{2}\right) = \left(0, \frac{h}{2}\right)\) and \(N = \left(\frac{a+b}{2}, \frac{0+h}{2}\right) = \left(\frac{a+b}{2}, \frac{h}{2}\right)\).
02

Show that MN is Parallel to AB

Since both \(M\) and \(N\) have the same y-coordinate, \(M N\) is on the line \(y = \frac{h}{2}\). Thus \(M N\) is parallel to \(AB\) since they both have the same y-coordinate and are horizontal.
03

Calculate the Length of MN

Using the midpoint coordinates, calculate the distance \(MN\) as follows:\[ MN = \sqrt{\left(\frac{a+b}{2} - 0\right)^2 + \left(\frac{h}{2} - \frac{h}{2}\right)^2} \]\[ = \frac{a+b}{2} \]. Thus, the length of \(MN\) is \(\frac{a+b}{2}\).
04

Find the Area of Trapezium ABCD

The area \(A\) of the trapezium is given by:\[ A = \frac{1}{2} \times (\text{Sum of parallel sides}) \times d \]\[ = \frac{1}{2} \times (a + b) \times d \].
05

Conclusion

We have shown that \(MN \parallel AB\) and its length is \(\frac{a+b}{2}\). The area of the trapezium is \(\frac{1}{2}(a+b)d\).

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Key Concepts

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

Trapezium
A trapezium, also known as a trapezoid in some regions, is a four-sided figure or quadrilateral with at least one pair of parallel sides. In the exercise, the sides \(AB\) and \(DC\) of trapezium \(ABCD\) are parallel to each other. Therefore, lines \(AB\) and \(DC\) are called the parallel sides or the "bases" of the trapezium.Trapeziums have diverse applications, including calculations of areas in geometric shapes and architectural designs. Understanding the properties of trapeziums, like the length of its sides and the height between the parallel sides, is fundamental for solving problems efficiently.
  • Parallel sides are crucial in defining the shape and characteristics of a trapezium.
  • The non-parallel sides are called the "legs".
  • It is necessary to identify the parallel sides to apply area formulas.
Coordinate Geometry
Coordinate geometry allows the representation of geometric figures through coordinates on the XY-plane. In this exercise, coordinate geometry aids us in determining the position of points \(A, B, C,\) and \(D\), and thus the midpoints \(M\) and \(N\), effortlessly.Assign coordinates to these points effectively helps in visualizing and solving geometrical problems. The provided example with coordinates demonstrates the power of using coordinate geometry to find properties like distances and parallelism.
  • Assigning coordinates simplifies complex geometric problems.
  • The method involves crucial concepts such as midpoints and slopes.
  • It integrates algebra with geometry to solve problems.
Midpoints
The midpoint of a line segment in geometry is the point that divides the segment into two equal parts. Finding the midpoint involves averaging the x-coordinates and y-coordinates of the endpoints.In the case of the trapezium \(ABCD\), we find that:
  • Midpoint \(M\) of \(AD\) is calculated as \( \left(0, \frac{h}{2}\right) \).
  • Midpoint \(N\) of \(BC\) is calculated as \( \left(\frac{a+b}{2}, \frac{h}{2}\right) \).
  • The process confirms that the midpoints lie on the same horizontal line, ensuring parallelism to the bases.
Midpoints are not just specific to segments within a trapezium; they are a general concept applicable to any line segment across various geometric shapes.
Parallel Lines
Parallel lines are lines in a plane that never meet; they are always the same distance apart. In a trapezium, identifying parallel sides is essential as they determine the nature of the shape. The exercise asks you to show that line \(MN\) is parallel to \(AB\), using the coincidence of their y-coordinates.Important characteristics of parallel lines include:
  • Parallel lines have equal distance between them at all points.
  • For horizontal or vertical lines, parallelism can be easily spotted by comparing slopes.
  • In practical scenarios, like constructions or map design, parallelism is used to keep elements aligned.
By understanding these characteristics, you can easily predict properties of shapes like trapeziums, which often need parallel lines for area and distance calculations.
Area Calculation
Calculating the area of a trapezium is straightforward once you know the lengths of the parallel sides and the height. From the solution, the formula used is: \[ A = \frac{1}{2} \times (\text{Sum of parallel sides}) \times d \]So, for trapezium \(ABCD\), the area is given by:\[ A = \frac{1}{2} \times (a + b) \times d \]Where:
  • \(a\) and \(b\) are lengths of the parallel sides.
  • \(d\) is the perpendicular distance between the parallel sides.
This formula is derived from the basic principle of decomposing the trapezium into simpler geometric shapes, making the area calculation both simple and reliable. Always ensure you have the correct lengths and height to apply this formula effectively.

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Most popular questions from this chapter

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 \(P Q .)\)

Let \(\triangle A B C\) be a general triangle on the unit sphere with the usual labelling (so that \(x\) denotes the length of the side of a triangle opposite vertex \(X,\) and \(X\) is used both to label the vertex and to denote the size of the angle at \(X)\). Prove that: $$\frac{\sin a}{\sin A}=\frac{\sin b}{\sin B}=\frac{\sin c}{\sin C}$$

(a) A regular \(n\) -gon is inscribed in a circle of radius \(r\). (i) Find the exact area \(a_{n}\) (in surd form): when \(n=3\); when \(n=4\); when \(n=5 ;\) when \(n=6 ;\) when \(n=8 ;\) when \(n=10 ;\) when \(n=12\). (ii) Check that, for each \(n\) : $$a_{n}=d_{n} \times r^{2}$$ for some constant \(d_{n},\) where $$d_{3}D_{4}>D_{5}>D_{6}>D_{8}>D_{10}>D_{12} \cdots$$ (c) Explain why \(d_{12}

Suppose that in \(\triangle A B C, \angle C=\angle A+\angle B\). Prove that \(C\) lies on the circle with diameter \(\underline{A B}\). (In particular, if the angles of \(\triangle A B C\) add to a straight angle, and \(\angle A C B\) is a right angle, then \(C\) lies on the circle with diameter \(\underline{A B}\).)

You are given two lines \(m\) and \(n\) crossing at the point \(B\). (a) If \(A\) lies on \(m\) and \(C\) lies on \(n,\) prove that each point \(X\) on the bisector of angle \(\angle A B C\) is equidistant from \(m\) and from \(n\). (b) If \(X\) is equidistant from \(m\) and from \(n,\) prove that \(X\) must lie on one of the bisectors of the two angles at \(B\).
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