Jeremy loves right triangles that have side length 1. In fact, he loves to combine them so that the hypotenuse of one becomes the leg of the next. He does this over and over to create beautiful pictures like this: [asy] pair A=(0,0),B=(0,1),C=(0.4404,1.957), D=(1.2,2.9), E=(3.316,0); draw(A--B--C--D--E--A); draw(B--E--C); draw(D--E); draw(rightanglemark(E,A,B)); draw(rightanglemark(E,B,C)); draw(rightanglemark(E,C,D)); label("1",midpoint(A--B),W); label("1",midpoint(C--B),NW); label("1",midpoint(C--D),NW); label("... (pattern continues here)",midpoint(D--E),E); [/asy] If the longer leg of the first triangle is $\sqrt{11}$, what will be the hypotenuse of the 2014th triangle?

Accepted Solution

Answer:   45Step-by-step explanation:The hypotenuse of the first triangle is ...   √((√11)² +1²) = √12 = √(n+11) . . . .  where n is the triangle numberThen the hypotenuse of the 2014th triangle will be ...   √(2014 +1) = √2015 = 45