In triangle ABC, AB = 10 and AC = 17. Let D be the foot of the perpendicular from A to BC.

a² + b² = c²

a² + b² = 10²

a² + b² = 17²

Using the info that it is a 3:5 ratio, you can determine that the maximium numbers it can be is 6:10 without being 10.

All other 3:5 ratios under 10 not using decimals are 3:5, since that would give AD = 16.7 and 8.6 which don't match up, so it isn't a whole number.

Next I used a simultaneus equation: 289 - y = 100 - x    and    3y = 5x.

What I got was x = 567/2 or 283.5 and y = 945/2 or 472.5

Squaring them gave me the answer: x = 16.8 and y = 21.7 which still didn't make sense.

So then I thought about it and it's impossible.

The question as written leads to a complex value for AD!!!!  Hoever, if BD/CD = 2:5 then AD has a more sensible value (=8).