geo met ry halp

How are you going to know how to do these on a test if you don't know the explanation??  

We can use the Pythagorean theorem to do this problem. The Pythagorean theorem says

a² + b²  =  c²      , where  " a "  and  " b "  are legs of a right triangle, and  " c " is the hypotenuse.

They tell us the legs are  264  and  495  .   So...

264² + 495²  =  c²              Multiply  264 * 264  to get  69,696  and  495 * 495  to get  245,025 .

69,696 + 245,025  =  c²     Add them

314,721  =  c²                    Take the positive square root of both sides.

561  =  c

Well, if I was truly not allowed to use a calculator during any step throughout this problem, I would do the following:
 

\(264^2=(200+60+4)(200+60+4)=200(200+60+4)+60(200+60+4)+4(200+60+4)\)

There are a lot of friendly numbers here, do you agree? Now, let's continue the expansion.

\(200(200+60+4)+60(200+60+4)+4(200+60+4)=40000+12000+800+12000+3600+240+800+240+16\)

WOAH! That's a lot of terms! I would first rearrange the numbers such that the largest are added to the smallest.

\(40000+12000+12000+3600+800+800+240+240+16\)

You can do addition in any order you'd like. The colors indicate an addition where the result will be friendly and will end in another 0.

Now, just do the addition. 

\(40000+12000+12000+3600+800+800+240+240+16=69696\)

I wouldn't do 495 the same way, however. I would break that into \((500-5)(500-5)\). Now, do the same thing like before.

\(500(500-5)-5(500-5)\)

\(250000-2500-2500+25\)

\(245025\)

The square root is where I would hit a barrier, though...

Note that   264  /  495  =    8  / 15

So......the ratio of the legs  is   8  :  15

And one of the Pythagorean "Triples"  is     8  : 15  : 17

So.........

264 / 8  =  33

And the hypotenuse must be......17 * 33  =   561

So......the  sides are    264 -  495  - 561