Question 1165102
An isosceles trapezium ABCD has perpendicular diagonals and side lengths
AB = 1 and CD = 7.
(a) Find the length of the two equal sides.
<pre>{{{drawing(600,400,-1,8,-1,5, 
line(0,0,7,0), line(0,0,3,4),line(3,4,4,4), line(4,4,7,0),
line(3,4,7,0), line(0,0,4,4),
locate(0,0,C),locate(7,0,D),locate(3,4.3,B),locate(4,4.3,A),
locate(3.47,3.4,E),locate(3.5,0,7), locate(3.47,4.3,1),
locate(1.94,2,a),locate(4.94,2,a),locate(3.3,3.9,b),locate(3.6,3.9,b)

 )}}}
Triangles ABE and CDE are similar right triangles, so
{{{a/7=b/1}}}
{{{a=7b}}}

The diagonals are perpendicular so we can use the Pythagorean theorem
on the right triangles.

{{{b^2+b^2=1^2}}}
{{{2b^2=1}}}
{{{b^2=1/2}}}
{{{b=sqrt(1/2)}}}
{{{b=1/sqrt(2)}}}
{{{b=sqrt(2)/2}}}

{{{a=7b=7sqrt(2)/2}}}

{{{BC^2=a^2+b^2}}}
{{{BC^2=(7sqrt(2)/2)^2+(sqrt(2)/2)^2}}}
{{{BC^2=(49*2)/4+2/4}}}
{{{BC^2=98/4+2/4}}}
{{{BC^2=100/4}}}
{{{BC^2=25}}}
{{{BC=5}}}

So the two equal sides are 5 each.</pre>
(b) Find the product of the lengths of the diagonals.<pre>
The two diagonals are equal, and they are a+b each

{{{a+b=sqrt(2)/2+7sqrt(2)/2=8sqrt(2)/2=4sqrt(2)}}}

Multiply them together, which means to square that:

{{{(4sqrt(2))^2=16*2=32}}}

That's the product of the diagonals.

Edwin</pre>