Lesson Trapezoids and their mid-lines

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<H2>Trapezoids and their mid-lines</H2>

In this lesson you will learn major definitions and facts related to &nbsp;trapezoids&nbsp; and their &nbsp;mid-lines.

Trapezoid &nbsp;is a quadrilateral which has two opposite sides parallel and the other two sides non-parallel &nbsp;(see &nbsp;Figure 1).

The parallel sides of a trapezoid are called its &nbsp;bases&nbsp; (sides &nbsp;AB&nbsp; and &nbsp;DC&nbsp; in &nbsp;Figure 1). 
The non-parallel sides of a trapezoid are called its &nbsp;lateral sides&nbsp; or &nbsp;legs&nbsp; (sides &nbsp;AD&nbsp; and &nbsp;BC&nbsp; in &nbsp;Figure 1).

Mid-line&nbsp; of a trapezoid is the line segment connecting the midpoints of the lateral sides of a trapezoid. 
The mid-line &nbsp;EF&nbsp; of the trapezoid &nbsp;ABCD&nbsp; is shown in &nbsp;Figure 2.

<TABLE> 
 <TR>
 <TD>
&nbsp;&nbsp;&nbsp;&nbsp;{{{drawing( 280, 160, 0.5, 7.5, 0.5, 4.5, 
 line( 1, 1, 7, 1), 
 line( 1, 1, 3, 4),
 line( 3, 4, 6, 4), 
 line( 7, 1, 6, 4),

locate ( 0.9, 1,   A),
            locate ( 6.9, 1,   B),
            locate ( 6.1, 4.3, C),
            locate ( 2.7, 4.3, D),

locate (4, 1.45, base),
            locate (4, 3.9,  base),

locate (6.4, 3.1,  lateral),
            locate (6.6, 2.7,  side),

locate (0.9, 3.1, lateral),
 locate (1.2, 2.7, side)
)}}}
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Figure 1. &nbsp;Trapezoid

</TD>
 <TD>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{drawing( 280, 160, 0.5, 7.5, 0.5, 4.5, 
 line( 1, 1, 7, 1), 
 line( 1, 1, 3, 4),
 line( 3, 4, 6, 4), 
 line( 7, 1, 6, 4),

locate ( 0.9, 1,   A),
            locate ( 6.9, 1,   B),
            locate ( 6.1, 4.3, C),
            locate ( 2.7, 4.3, D),

line( 2.0, 2.5, 6.5, 2.5),
 locate ( 1.7, 2.7, E),
 locate ( 6.6, 2.7, F)
)}}}
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Figure 2. &nbsp;Trapezoid and its mid-line

</TD>
 </TR>
</TABLE>

<H3>Theorem 1</H3>The mid-line of a trapezoid is parallel to its bases. 
The length of the mid-line of a trapezoid is half of the sum of the lengths of its bases.

<TABLE> 
 <TR>
 <TD>
Proof

Let &nbsp;ABCD&nbsp; be a trapezoid with the bases &nbsp;AB&nbsp; and &nbsp;DC&nbsp; and the mid-line &nbsp;EF&nbsp;

(Figure 2). &nbsp;Let us draw the straight line &nbsp;DF&nbsp; through the points &nbsp;D&nbsp; and &nbsp;F

till the intersection with the extension of the straight line &nbsp;AB&nbsp; at the point

G&nbsp; (Figure 3). &nbsp;Compare the triangles &nbsp;DFC&nbsp; and &nbsp;FBG.

The segments &nbsp;FC&nbsp; and &nbsp;BF&nbsp; are congruent since the point &nbsp;F&nbsp; is the midpoint

of the side &nbsp;BC. &nbsp;The angles &nbsp;DFC&nbsp; and &nbsp;BFG are congruent as the vertical angles.

The angles &nbsp;DCF&nbsp; and &nbsp;FBG are congruent as the alternate exterior angles

at the parallel lines &nbsp;AB&nbsp; and &nbsp;DC&nbsp; and the transverse &nbsp;BC&nbsp; (see the lesson

<A HREF=http://www.algebra.com/algebra/homework/Angles/Parallel-lines.lesson>Parallel lines</A>&nbsp; under the topic &nbsp;Angles, complementary, supplementary angles

</TD>
 <TD>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{drawing( 400, 160, 0.5, 10.5, 0.5, 4.5, 
 line( 1, 1, 7, 1), 
 line( 1, 1, 3, 4),
 line( 3, 4, 6, 4), 
 line( 7, 1, 6, 4),

locate ( 0.9, 1,   A),
            locate ( 6.9, 1,   B),
            locate ( 6.1, 4.3, C),
            locate ( 2.7, 4.3, D),

line( 2.0, 2.5, 6.5, 2.5),
            locate ( 1.7, 2.8, E),
            locate ( 6.6, 2.8, F),

green(line( 3, 4, 10, 1)),
            line( 7, 1, 10, 1),
            locate ( 10.0, 1, G),

line( 6.6, 1.7, 6.8, 1.7), 
            line( 6.6, 1.8, 6.8, 1.8),

line( 6.1, 3.3, 6.3, 3.3), 
            line( 6.1, 3.4, 6.3, 3.4),

arc( 6.5, 2.5, 0.8, 0.8, 27, 75),
             arc( 6.5, 2.5, 1.0, 1.0, 27, 75),

arc( 6.5, 2.5, 0.8, 0.8, 207, 252),
             arc( 6.5, 2.5, 1.0, 1.0, 207, 252),

arc( 7.0, 1.0, 0.8, 0.8, 255, 360),

arc( 6.0, 4.0, 0.8, 0.8, 75, 180),

locate (4.4, 1.45, a),
 locate (4.4, 4.45, d),
 locate (8.3, 1.45, d)
)}}}
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Figure 3. &nbsp;To the proof of the Theorem 1

</TD>
 </TR>
</TABLE>of the section &nbsp;Geometry&nbsp; in this site).

Hence, the triangles &nbsp;DFC&nbsp; and &nbsp;FBG are congruent in accordance with the &nbsp;ASA-test of congruency of triangles &nbsp;(see the &nbsp;Postulate 2&nbsp; 
of the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Triangles/Congruence-tests-for-triangles.lesson>Congruence tests for triangles</A>&nbsp; under the topic &nbsp;Triangles&nbsp; of the section &nbsp;Geometry&nbsp; in this site).

It implies that the segments &nbsp;DF&nbsp; and &nbsp;GF&nbsp; are congruent as the corresponding sides of the congruent triangles &nbsp;DFC&nbsp; and &nbsp;FBG.

Thus the mid-line &nbsp;EF&nbsp; of the trapezoid &nbsp;ABCD&nbsp; is the straight line segment connecting the midpoints of the triangle &nbsp;AGD.

It is well known fact that the the straight line segment connecting the midpoints of the triangle &nbsp;AGD&nbsp; is parallel to the triangle base &nbsp;AG&nbsp; and its length is half of the length of the triangle base. &nbsp;See the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Triangles/The-line-segment-joining-the-midpoints-of-two-sides-of-a-triangle.lesson>The line segment joining the midpoints of two sides of a triangle</A>&nbsp; under the topic &nbsp;Triangles&nbsp; of the section &nbsp;Geometry&nbsp; in this site.

In our case, &nbsp;the length of the segment &nbsp;EF&nbsp; is half of the length &nbsp;AG&nbsp;: &nbsp;&nbsp;|EF| = {{{1/2}}}*|AG| = {{{1/2}}}*(|AB| + |BG|).

Since &nbsp;|BG| = |DC|&nbsp; from the triangles congruency, &nbsp;we have &nbsp;|EF| = {{{1/2}}}*(|AB| + |DC|), &nbsp;or &nbsp;|EF| = {{{1/2}}}*(a + d), &nbsp;where &nbsp;a&nbsp; and &nbsp;d&nbsp; are the lengths of the trapezoid bases.

Thus the proof of the &nbsp;Theorem 1&nbsp; is fully completed.

<H3>Theorem 2</H3>In a trapezoid, &nbsp;the line segment drawn from the midpoint of the lateral side parallel to the bases intersects the other lateral side at its midpoint.

<TABLE> 
 <TR>
 <TD>
Proof

The proof is very close to that of the Theorem 1 above.

Let &nbsp;ABCD&nbsp; be a trapezoid with the bases &nbsp;AB&nbsp; and &nbsp;CD, &nbsp;and let &nbsp;EF&nbsp; be the

straight line drawn through the midpoint &nbsp;E&nbsp; of the lateral side &nbsp;AD&nbsp; parallel

to the bases &nbsp;(Figure 4). &nbsp;Let us draw the straight line &nbsp;DF&nbsp; through the

points &nbsp;D&nbsp; and &nbsp;F till the intersection with the extension of the straight line

&nbsp;AB&nbsp; at the point G&nbsp; (Figure 4). &nbsp;Compare the triangles &nbsp;DFC&nbsp; and &nbsp;FBG.

Since the straight line &nbsp;EF&nbsp; passes through the midpoint &nbsp;E&nbsp; of the triangle

AGD&nbsp; and is parallel to its base &nbsp;AB, &nbsp;it intersects &nbsp;the other side &nbspGD&nbsp; of

the triangle at its midpoint &nbsp;F&nbsp; too &nbsp;(see the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Triangles/The-line-segment-joining-the-midpoints-of-two-sides-of-a-triangle.lesson>The line segment

joining the midpoints of two sides of a triangle</A>&nbsp; under the topic &nbsp;Triangles

of the section &nbsp;Geometry&nbsp; in this site). &nbsp;Hence, &nbsp;the segments &nbspDF&nbsp; and &nbspFG&nbsp;

</TD>
 <TD>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{drawing( 400, 160, 0.5, 10.5, 0.5, 4.5, 
 line( 1, 1, 7, 1), 
 line( 1, 1, 3, 4),
 line( 3, 4, 6, 4), 
 line( 7, 1, 6, 4),

locate ( 0.9, 1,   A),
            locate ( 6.9, 1,   B),
            locate ( 6.1, 4.3, C),
            locate ( 2.7, 4.3, D),

line( 2.0, 2.5, 6.5, 2.5),
            locate ( 1.7, 2.8, E),
            locate ( 6.6, 2.8, F),

green(line( 3, 4, 10, 1)),
            line( 7, 1, 10, 1),
            locate ( 10.0, 1, G),

green(line( 4.45, 3.18, 4.95, 3.18)), 
      green(line( 4.45, 3.32, 4.95, 3.32)),

green(line( 7.95, 1.84, 8.45, 1.84)), 
      green(line( 7.95, 1.72, 8.45, 1.72)),

arc( 6.5, 2.5, 0.8, 0.8, 27, 75),
             arc( 6.5, 2.5, 1.0, 1.0, 27, 75),

arc( 6.5, 2.5, 0.8, 0.8, 207, 252),
             arc( 6.5, 2.5, 1.0, 1.0, 207, 252),

arc( 10.0, 1.0, 1.2, 1.2, 180, 207),

arc( 3.0, 4.0, 1.2, 1.2, 0, 25)
)}}}
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Figure 4. &nbsp;To the proof of the Theorem 2

</TD>
 </TR>
</TABLE>are congruent. Thus the triangles &nbsp;DFC&nbsp; and &nbsp;FBG&nbsp; have the congruent sides &nbsp;DF&nbsp; and &nbsp;FG.

Further, &nbsp;the angles &nbsp;DFC&nbsp; and &nbsp;BFG are congruent as the vertical angles.

The angles &nbsp;FDC&nbsp; and &nbsp;FGB are congruent as the alternate interior angles at the parallel lines &nbsp;AG&nbsp; and &nbsp;DC&nbsp; and the transverse &nbsp;DG&nbsp; 
(see the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Angles/Parallel-lines.lesson>Parallel lines</A>&nbsp; under the topic &nbsp;Angles, complementary, supplementary angles&nbsp; of the section &nbsp;Geometry&nbsp; in this site).

It implies that the segments &nbsp;CF&nbsp; and &nbsp;FB&nbsp; are congruent as the corresponding sides of the congruent triangles &nbsp;DFC&nbsp; and &nbsp;FBG.

Thus the point &nbsp;F&nbsp; is the midpoint of the side &nbsp;BC&nbsp; of the trapezoid &nbsp;ABCD. &nbsp;This is what has to be proved.

<H3>Summary</H3>1. &nbsp;The mid-line of a trapezoid is parallel to its bases. 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The length of the mid-line of a trapezoid is half of the sum of the lengths of its bases.

2. &nbsp;In a trapezoid, &nbsp;the line segment drawn from the midpoint of the lateral side parallel to the bases intersects the other lateral side at its midpoint.

<H3>Example 1</H3>In a trapezoid the bases are of &nbsp;17 cm&nbsp; and &nbsp;13 cm. &nbsp;Find the length of the trapezoid's mid-line.

Solution

The mid-line of the trapezoid is half of the sum of the lengths of its bases, &nbsp;i.e. &nbsp;{{{(13 + 17)/2}}} = {{{30/2}}} = 15 cm.

Answer. &nbsp;The length of the trapezoid's mid-line is &nbsp;15 cm.

<H3>Example 2</H3>In a trapezoid, &nbsp;the larger base is of &nbsp;27 cm long, &nbsp;and it is in &nbsp;10 cm&nbsp; longer than the shorter base. &nbsp;Find the length of the trapezoid's mid-line.

Solution

The shorter base length is &nbsp;27 cm - 10 cm = 17 cm. 
The mid-line of the trapezoid is half of the sum of the lengths of its bases, &nbsp;i.e. &nbsp;{{{(17 + 27)/2}}} = {{{44/2}}} = 22 cm.

Answer. &nbsp;The length of the trapezoid's mid-line is &nbsp;22 cm.

<H3>Example 3</H3>In a trapezoid, &nbsp;the larger base is in &nbsp;10 cm longer than the shorter base, &nbsp;and its mid-line is of &nbsp;22 cm&nbsp; long. &nbsp;Find the lengths of the trapezoid's bases.

Solution

Let x be the length of the larger base of the trapezoid in centimeters. 
Then the shorter base length is &nbsp;(x-10) cm. 
Since the mid-line of the trapezoid is half of the sum of the lengths of its bases, &nbsp;i.e. &nbsp;{{{(x + (x-10))/2}}} cm, &nbsp;it gives the equation

{{{(x + (x-10))/2}}} = 22.

Simplify this equation step by step and get the solution:

{{{(2x - 10)/2}}} = 22,

{{{2x - 10}}} = {{{44}}},

2x = 54,

x = 27.

So, &nbsp;the length of the larger base is &nbsp;27 cm. &nbsp;The length of the shorter base is &nbsp;27 cm - 10 cm = 17 cm.

Answer. &nbsp;The length of the trapezoid's larger base is &nbsp;27 cm, &nbsp;the length of the shorter base is &nbsp;17 cm.

My other lessons on trapezoids in this site are

&nbsp;&nbsp;&nbsp;&nbsp;-<A HREF =http://www.algebra.com/algebra/homework/Polygons/Trapezoids-and-their-base-angles.lesson>Trapezoids and their base angles</A>,

&nbsp;&nbsp;&nbsp;&nbsp;-<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>,

&nbsp;&nbsp;&nbsp;&nbsp;-<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>,

&nbsp;&nbsp;&nbsp;&nbsp;-<A HREF =http://www.algebra.com/algebra/homework/Polygons/Solving-problems-on-trapezoids.lesson>Solving problems on trapezoids</A>,

&nbsp;&nbsp;&nbsp;&nbsp;-<A HREF =http://www.algebra.com/algebra/homework/Polygons/Solving-problems-on-isoscales-trapezoids.lesson>Solving problems on isosceles trapezoids</A>,

&nbsp;&nbsp;&nbsp;&nbsp;-<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>,

&nbsp;&nbsp;&nbsp;&nbsp;-<A HREF =http://www.algebra.com/algebra/homework/Polygons/HOW-TO-construct-a-trapezoid-given-by-the-lengths-of-its-sides.lesson>HOW TO construct a trapezoid given by the lengths of its sides</A> &nbsp;&nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;-<A HREF =http://www.algebra.com/algebra/homework/Polygons/PROPERTIES-OF-TRAPEZOIDS.lesson>PROPERTIES OF TRAPEZOIDS</A>

under the current topic, &nbsp;&nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;-<A HREF =http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-trapezoids.lesson>Solved problems on trapezoids</A> &nbsp;&nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;-<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 &nbsp;Geometry&nbsp; of the section &nbsp;Word problems.

To navigate over all topics/lessons of the Online Geometry Textbook use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.