Lesson HOW TO construct a trapezoid given by the lengths of its sides
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<H2>HOW TO construct a trapezoid given by the lengths of its sides</H2> In this lesson you will learn how to construct a trapezoid using the ruler and the compass, if the lengths of its bases and the lengths of its lateral sides are given. The lesson is a continuation of the lesson <A HREF=http://www.algebra.com/algebra/homework/Polygons/Trapezoid-is-uniquely-defined-by-its-sides.lesson>Trapezoid is uniquely defined by the lengths of its sides</A> under the current topic. <H3>Problem 1</H3>Using the ruler and the compass, construct a trapezoid which has the bases congruent to the two given segments and the lateral sides congruent to the two other given segments. <BLOCKQUOTE>It is assumed that the four given segments satisfy all necessary conditions in order such a trapezoid could exist. You may think that the trapezoid did exist with the given bases and the lateral sides and was de-assembled into separate segments, so that now you need to construct (to restore) it again using the ruler and the compass.</BLOCKQUOTE> Note that if the trapezoid with the given sides does exist, then it is unique, according to the <B>Problem 1</B> of the lesson <A HREF=http://www.algebra.com/algebra/homework/Polygons/Trapezoid-is-uniquely-defined-by-its-sides.lesson>Trapezoid is uniquely defined by the lengths of its sides</A>. The solution of the <B>Problem 1</B> of that lesson makes a tip to constructing the trapezoid. <B>Construction</B> <TABLE> <TR> <TD> Let the two given segments <B>a</B> and <B>d</B> be the bases of the trapezoid, and the two other segments <B>b</B> and <B>c</B> be its lateral sides. First, construct the segment <B>e</B> congruent to the difference of the larger segment <B>a</B> and the shorter segment <B>d</B>: <B>e</B> = <B>a</B> - <B>d</B>. It is clear how to do it using the ruler and the compass. Second, construct the triangle given by its three sides <B>b</B>, <B>c</B> and <B>e</B>. (<B>Figure 1</B>). The method of constructing a triangle given by its three sides using a compass and a ruler is described in the lesson <A HREF=http://www.algebra.com/algebra/homework/Triangles/How-to-draw-a-congruent-triangle-using-a-compass-and-a-ruler.lesson>HOW TO construct a triangle using a compass and a ruler</A> in this site. </TD> <TD> {{{drawing( 560, 160, 0.5, 14.5, 0.5, 4.5, line( 1, 1, 4, 1), line( 1, 1, 3, 4), line( 3, 4, 4, 1), locate ( 0.8, 1, A), locate ( 2.8, 4.3, D), line( 8, 1, 14, 1), line( 8, 1, 10, 4), line( 10, 4, 13, 4), line( 14, 1, 13, 4), locate ( 7.8, 1, A), locate ( 13.9, 1, B), locate ( 13.1, 4.3, C), locate ( 9.8, 4.3, D), line( 3, 4, 4, 1), locate ( 3.9, 1, E), green(line( 10, 4, 11, 1)), locate ( 10.9, 1, E), locate ( 1.7, 2.8, b), locate ( 2.2, 1.4, a-d), locate ( 3.6, 2.8, b), locate ( 8.7, 2.8, b), locate ( 9.2, 1.4, a-d), locate ( 10.6, 2.8, b), locate ( 13.6, 2.8, b), locate ( 11.6, 4.4, d), locate ( 11.6, 1.4, a), locate ( 0.8, 4.0, Step_2), locate ( 7.8, 4.0, Step_3) )}}} <B>Figure 1</B>. Constructing the trapezoid <B>ABCD</B> (Step 2 and Step 3) </TD> </TR> </TABLE> The last, third step is to construct the trapezoid <B>EBCD</B> with the opposite sides <B>d</B> and <B>c</B>, which is attached to the triangle <B>AED</B> at the side <B>DE</B> (<B>Figure 1</B>). To do it, you need to extend the straight line segment <B>AE</B> to the segment <B>AB</B> by adding the segments <B>a-d</B> and <B>d</B>, which is elementary, and then to construct the straight line segment <B>DC</B> = <B>d</B> passing through the point <B>D</B> parallel to the straight line <B>AB</B>. It is described in the lesson <A HREF=http://www.algebra.com/algebra/homework/Triangles/How-to-draw-a-parallel-line-passing-through-a-given-point-using-a-compass-and-a-ruler.lesson>How to draw a parallel line passing through a given point using a compass and a ruler</A> in this site how to construct a straight line parallel to the given straight line and passing through the given point using the ruler and the compass. The construction is completed. It is clear from the construction procedure that we got the trapezoid with the given bases <B>a</B> and <B>d</B> and the given lateral sides <B>b</B> and <B>c</B> as required. My other lessons on trapezoids in this site are -<A HREF=http://www.algebra.com/algebra/homework/Polygons/Trapezoids-and-their-base-angles.lesson>Trapezoids and their base angles</A> -<A HREF =http://www.algebra.com/algebra/homework/Polygons/Trapesoids-and-their-mid-lines.lesson>Trapezoids and their mid-lines</A> -<A HREF =http://www.algebra.com/algebra/homework/Polygons/Diagonals-of-an-isosceles-trapezoid-are-congruent.lesson>Diagonals of an isosceles trapezoid are congruent</A> -<A HREF =http://www.algebra.com/algebra/homework/Polygons/Mid-line-of-a-trapezoid-is-the-locus-of-points-equidistant-from-its-bases.lesson>Mid-line of a trapezoid is the locus of points equidistant from its bases</A> -<A HREF =http://www.algebra.com/algebra/homework/Polygons/Solving-problems-on-trapezoids.lesson>Solving problems on trapezoids</A> -<A HREF =http://www.algebra.com/algebra/homework/Polygons/Solving-problems-on-isoscales-trapezoids.lesson>Solving problems on isosceles trapezoids</A> -<A HREF =http://www.algebra.com/algebra/homework/Polygons/Trapezoid-is-uniquely-defined-by-its-sides.lesson>Trapezoid is uniquely defined by the lengths of its sides</A> under the current topic, and -<A HREF =http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-trapezoids.lesson>Solved problems on trapezoids</A> -<A HREF =http://www.algebra.com/algebra/homework/word/geometry/Solving-problems-on-isosceles-trapezoids.lesson>Solved problems on isosceles trapezoids</A> under the topic <B>Geometry</B> of the section <B>Word problems</B>. To navigate over all topics/lessons of the Online Geometry Textbook use this file/link <A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.
