We have three similar triangles, because each has a right angle and shares an angle. Let's write the angles in order: opposite to short leg, long leg, hypotenuse.
CAB similar to BAD similar to CBD
Or as ratios,
CA:AB:CB = BA:AD:BD = CB:BD:CD
We also know
AC = AD + CD
(AD+CD):AB:CB = BA:AD:BD = CB:BD:CD
(AD+CD)/CB=CB/CD
We have CB=6, AD=5 and seek x=CD.
[tex](5 + x)/6 = 6/x[/tex]
[tex]x(x+5) = 36[/tex]
[tex]x^2 +5x - 36 = 0[/tex]
[tex](x+9)(x-4) = 0[/tex]
We reject the negative root and conclude x=4
Answer: 4
