The hypotenuse of a right triangle is 2 ft. less than three times the shorter leg. The longer leg is 3 ft. less than three times the shorter leg. Find the length of the shorter leg.
Let
c -----> the hypotenuse
a ----> the shorter leg
b ----> the longer leg
we have that
c=3a-2 -----> equation 1
b=3a-3 -----> quation 2
Applying the Pythagaroean Theorem
c^2=a^2+b^2
substitute
the values
(3a-2)^2=a^2+(3a-3)^2
solve for a
9a^2-12a+4=a^2+9a^2-18a+9
simplify
-12a+4=a^2-18a+9
a^2-18a+12a+9-4=0
a^2-6a+5=0
Solve teh quadratic equation
Solve by graphing
the solution is a=1 and a=5
see the attached figure
please wait a minute
Verify each solution of a
For a=1
c=3a-2 ------> c=3(1)-2=1
Is not a solution because c is the greater side
For a=5
c=3(5)-2=13
b=3(5)-3=12
therefore
the answer is
the shorter leg is 5 units