Question 1129618
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13 = 9 + 4 = {{{3^2}}} + {{{2^2}}}.


Therefore,   {{{sqrt(13)}}}  is the length of the hypotenuse of the right angled triangle with the legs of  3  and  2  centimeters long.



<pre>
    So, with a liner and a compass the construction is as follows.


    Draw a straight line.


    From the point A on the line, construct the segment AB on the line of the length 3 centimeters.


    Construct the perpendicular line to the straight line AB at the point B.

    On the perpendicular, construct the segment BC of the length 2 cm.

    Connect the point A and C by the liner.


    The segment AC is what you need.
</pre>

Solved.


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On basic operations in construction problems with a liner and a compass, see the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Triangles/How-to-bisect-a-segment-using-a-compass-and-a-ruler.lesson>HOW TO bisect a segment using a compass and a ruler</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Triangles/-HOW-TO-bisect-an-angle-using-a-compass-and-a-ruler.lesson>HOW TO bisect an angle using a compass and a ruler</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://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.


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Also, &nbsp;you have this free of charge online textbook on Geometry

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A> 

in this site.


The referred lessons are the part of this online textbook under the topic &nbsp;"<U>Geometric constructions using a compass and a ruler. Basic operations</U>".



Save the link to this online textbook together with its description


Free of charge online textbook in GEOMETRY
https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson


to your archive and use it when it is needed.


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Notice that the construction algorithm, given in the post by &nbsp;@josmicely, &nbsp;gives you the segment of the length  &nbsp;{{{13/sqrt(2)}}}, 
which is &nbsp;<U>totally different</U> &nbsp;from what the problem does require.