What is the area of triangle whose sides are sqrt(61), sqrt(153), sqrt(388)

You get partial credit for solving it with Heron's Formula, and full (or even extra) credit for using the following hint: each of the numbers 61, 153, and 388 is the product of a square and one or more primes of the form 4k+1.

Source: Henry Dudeney

The area is 21.

Primes of the form 4k+1 can be expressed as the sum of two squares.  And the product of two numbers, each expressed as the sum of two squares, can also be expressed as the sum of two squares.  Together, these two facts add up to the following important theorem: Any number that can be expressed as the product of a square and one or more primes of the form 4k+1 can be expressed as the sum of two squares.  The proof, and a procedure for finding the squares, is on this page.  The procedure is fairly hard to grasp, I think, so for small numbers like these, it's easier to do some "trial and error".  First note that:

61 is a prime of the form 4k+1,
153 is the product of 9 (a square) and 17, which is a prime of the form 4k+1, and
388 is the product of 4 (a square), and 97, which, too, is a prime of the form 4k+1.

Now try adding up some squares to get 61, 17, and 97.  You should quickly see that:

61 = 6²+5²,
17 = 4²+1², so 3²*17 = 153 = 12²+3², and
97 = 9²+4², so 2²*97 = 388 = 18² + 8².

Also, note that the two numbers whose squares add up to 61 (6 and 5) can be added pairwise with the two numbers whose squares add up to 153 (12 and 3) to form the two numbers whose squares add up to 388 (18 and 8).  By combining this information, we see that we can draw the following diagram, showing each of the three sides of the original triangle as a hypotenuse of a right triangle.  Moreover, since the legs of the three triangles add up nicely (5+3=8 and 6+12=18), the corresponding legs are all parallel to one another:

Now the area of the largest right triangle minus the areas of the two smaller right triangles minus the area of the rectangle equals the area of the original triangle:

(1/2)(8)(18) - (1/2)(5)(6) - (1/2)(3)(12) - (3)(6) = 21