Question 837675
{{{drawing(400,200,-11,18,-1,13.5,
green(rectangle(0,0,1.5,1.5)),
triangle(-9,0,16,0,0,12),
line(-0.9,10.8,0.3,9.9),line(1.2,11.1,0.3,9.9),
green(line(0,0,0,12)),
locate(-4.7,1,9),locate(7.7,1,16),
locate(-9.6,0.5,A),locate(16.1,0.5,C),
locate(-0.3,13,B)
)}}}
The altitude BD splits right triangle ABC into right triangles ADB and BCD.
They are all similar right triangles.
The ratio {leg adjacent to angle BAC)/hypotenuse is
{{{AB/AC=AB/(AD+DC)}}} in triangle ABC,
and it is {{{AD/AB}}} in triangle ADB.
Since those two triangles are similar, those ratios are the same,
so {{{AB/(AD+DC)=AD/AB}}} <--> {{{(AB)^2=AD*(AD+DC)}}} .
Substituting numbers,
{{{(AB)^2=9(9+16)}}}
{{{(AB)^2=9*25}}}
{{{(AB)^2=3^2*5^2}}}
{{{(AB)^2=(3*5)^2}}}
{{{(AB)^2=15^2}}}
{{{highlight(AB=15)}}}