625
/www/nrich/html/content/99/02/six2/
1
2011-02-01T00:00:01
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">
<mdo:image src="linkage.gif" alt=""></mdo:image> <br></br>
<p>Four rods, two of length <em>a</em> and two of length
<em>b</em>, are linked to form a kite, as shown in the diagram. The
linkage is moveable so that the angles change. What is the maximum
area of the kite?</p>
<p>Now suppose the four rods are assembled into a linkage which
makes a parallelogram. What is the maximum area of this
parallelogram?</p>
</mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p><span class="editorial">Congratulations Christian,Nisha and Suzann, The Mount School York for seeing just how simple this problem really is - just two right angled triangles. Well done also to Sarah and Alice from the same school; they used</span> 'half <em>ab</em> sin <strong>C</strong> ' <span class="editorial">but</span> 'half base times height' <span class="editorial">is really all that is
needed.</span></p>
<p>The kite is made up of two triangles if we divide it along it's axis of symmetry.</p>
<p>If we call <em>b</em> the base of the triangle then its maximum height has got to be <em>a</em> , see the diagram below.</p>
<div style="text-align: center;"><mdo:image alt="" height="274" src="linkage_s.gif" width="237"></mdo:image></div>
<br></br>
<p>The area of this triangle is:</p>
<table style="" border="">
<tbody>
<tr>
<td>
<table style="" border="">
<tbody>
<tr>
<td align="center" nowrap="nowrap" style=""><em>A</em> =</td>
<td align="center" nowrap="nowrap" style=""><em>b</em> x <em>h</em>
<hr noshade="noshade"></hr>
2</td>
<td align="center" nowrap="nowrap" style=""> </td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
<br></br>
<p>and <em>h</em> will always be smaller than <em>a</em> as <em>a</em> is the hypotenuse of the right-angled triangle made up of the green lines and side <em>a</em> .</p>
<p>The maximum value of the area of the triangle will be when <em>h</em> = <em>a</em> , so the maximum area of the triangle is :</p>
<table style="" border="">
<tbody>
<tr>
<td>
<table style="" border="">
<tbody>
<tr>
<td align="center" nowrap="nowrap" style=""><em>A</em> =</td>
<td align="center" nowrap="nowrap" style=""><em>b</em> x <em>a</em>
<hr noshade="noshade"></hr>
2</td>
<td align="center" nowrap="nowrap" style=""> </td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
<br></br>
<p>and the area of the kite will just be <em>b</em> x <em>a</em></p>
<p>If the rods are made into a parallelogram it won't change things because the parallelogram is made up of the same two triangles as the kite.<br></br>
<br></br>
<span class="editorial">This means that the parallelogram of maximal area is actually a rectangle, well done!</span></p>
<br></br></mdoxml>
2
5
0
0
1
0
0
Linkage
Four rods, two of length a and two of length b, are linked to form
a kite. The linkage is moveable so that the angles change. What is
the maximum area of the kite?
2D Geometry, Shape and Space
Parallelograms
2D Geometry, Shape and Space
Mixed triangles
2D Geometry, Shape and Space
Quadrilaterals
Using, Applying and Reasoning about Mathematics
Visualising