The hypotenuse of a right triangle is 39 inches, and the length of one leg is 6 inches longer than twice the other leg. How do you find the length of each leg?

1 Answer
Nov 14, 2016

Answer:

The legs are of length #15# and #36#

Explanation:

Method 1 - Familiar triangles

The first few right angled triangles with an odd length side are:

#3, 4, 5#

#5, 12, 13#

#7, 24, 25#

Notice that #39 = 3 * 13#, so will a triangle with the following sides work:

#15, 36, 39#

i.e. #3# times larger than a #5, 12, 13# triangle ?

Twice #15# is #30#, plus #6# is #36# - Yes.

#color(white)()#
Method 2 - Pythagoras formula and a little algebra

If the smaller leg is of length #x#, then the larger leg is of length #2x+6# and the hypotenuse is:

#39 = sqrt(x^2 + (2x+6)^2)#

#color(white)(39) = sqrt(5x^2+24x+36)#

Square both ends to get:

#1521 = 5x^2+24x+36#

Subtract #1521# from both sides to get:

#0 = 5x^2+24x-1485#

Multiply both sides by #5# to get:

#0 = 25x^2+120x-7425#

#color(white)(0) = (5x+12)^2-144-7425#

#color(white)(0) = (5x+12)^2-7569#

#color(white)(0) = (5x+12)^2-87^2#

#color(white)(0) = ((5x+12)-87)((5x+12)+87)#

#color(white)(0) = (5x-75)(5x+99)#

#color(white)(0) = 5(x-15)(5x+99)#

Hence #x = 15# or #x = -99/5#

Discard the negative solution since we are seeking the length of the side of a triangle.

Hence the smallest leg is of length #15# and the other is #2*15+6 = 36#