Question 89044
We basically have this triangle set up:


{{{drawing(500,500,-0.5,3,-0.5,3,

line(0,0,0,3),
line(0,3,2,0),
line(2,0,0,0),
locate(-0.2,1.5,a),
locate(1,-0.2,15),
locate(1,2,c)
)}}}



{{{a^2+15^2=c^2}}} Here is the original Pythagorean equation



{{{drawing(500,500,-0.5,5,-0.5,5,

blue(line(0,0,0,3)),
blue(line(0,3,2,0)),
blue(line(2,0,0,0)),

green(line(0,0,0,1)),
green(line(0,1,4,0)),
green(line(4,0,0,0)),
locate(-0.2,1.5,a),
locate(-0.2,0.5,a-13),
locate(1,-0.2,15),
locate(2.2,-0.2,15+9),
locate(1,2,c),
locate(2.5,0.6,c)
)}}} Here is the original position of the ladder (blue) and the shifted ladder (green)

Notice that we subtract 13 from a to get {{{a-13}}}. So our new "a" is {{{a-13}}}. Also, we add 9 to our base to get {{{15+9}}}. So our new "b" is {{{15+9=24}}}


{{{(a-13)^2+24^2=c^2}}} So if we shift the ladder, we get this equation. 



Notice how the 2 equations are equal to {{{c^2}}}, so that means we can set the two equations equal to each other


{{{(a-13)^2+24^2=a^2+15^2}}}


{{{(a-13)^2+576=a^2+225}}} Evaluate {{{24^2}}} and {{{15^2}}}


{{{a^2-26a+169+576=a^2+225}}} Foil {{{(a-13)^2}}}


{{{a^2-26a+745=a^2-520}}} Combine like terms


{{{-26a+745=-520}}} Subtract {{{a^2}}} from both sides


{{{-26a=-520}}} Subtract 745 from both sides


{{{a=20}}} Divide both sides by -26


So the original height of the ladder is 20 feet, and it drops to {{{20-13=7}}} feet. Lets check our answer:


Check:


{{{20^2+15^2=400+225=625}}}



{{{7^2+24^2=49+576=625}}}


Since the 2 equations are equal, this means that we get the same hypotenuse. So the ladder is {{{sqrt(625)=25}}} feet long. This also verifies our answer.