SOLUTION: A right triangle has sides of integer length and perimeter 2010 units. The length of the shortest side is equal to one-fifth of the sum of the lengths of the other two sides. What

Algebra ->  Pythagorean-theorem -> SOLUTION: A right triangle has sides of integer length and perimeter 2010 units. The length of the shortest side is equal to one-fifth of the sum of the lengths of the other two sides. What       Log On


   

Question 266509: A right triangle has sides of integer length and perimeter 2010 units. The length of the shortest side is equal to one-fifth of the sum of the lengths of the other two sides. What is the area of the triangle?
Found 2 solutions by Edwin McCravy, vksarvepalli:
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
A right triangle has sides of integer length and perimeter 2010 units. The length of the shortest side is equal to one-fifth of the sum of the lengths of the other two sides. What is the area of the triangle? 

It's a matter of solving this system of equations
and substituting in the right triangle area formula
A=%281%2F2%29ab

system%28a%2Bb%2Bc=2010%2C%0D%0Aa%5E2%2Bb%5E2=c%5E2%2C%0D%0Aa=%281%2F5%29%28b%2Bc%29%29

Clear fractions in the third equation:

5a=b%2Bc

Solve the first equation for b%2Bc%29

b%2Bc=2010-a

Therefore

5a=2010-a

4a=2010

a=335 

Substituting in 

a%2Bb%2Bc=2010

335%2Bb%2Bc=2010

Solve for c

c=1675-b

Substitute that and a=335
in the Pythagorean equation

a%5E2%2Bb%5E2=c%5E2

335%5E2%2Bb%5E2=%281675-b%29%5E2

112225%2Bb%5E2=2805625+-+3350b+%2B+b%5E2

Cancel the b%5E2's

112225%2Bb%5E2=2805625+-+3350b+%2B+b%5E2

-26693400=-3350b

3350b=26693400

b=804

It isn't necessary to find c but we'll do it anyway

so we can check if we like.

Substitute that in

c=1675-b

c=1675-804

c=871

But all we want is the area, so

we substitute a=335 and b=804

A=%281%2F2%29ab

A=%281%2F2%29%28335%29%28804%29

A = 134,670 square units.

It was not necessary to tell us that the sides are integers.

Edwin

Answer by vksarvepalli(154) About Me  (Show Source):
You can put this solution on YOUR website!
let the sides be a,b & c with a being the shortest and c the hypotenuse(longest)
the are of the triangle will be 1/2*a*b (half*base*height)
given
The length of the shortest side is equal to one-fifth of the sum of the lengths of the other two sides
so a=1/5 *(b+c)
=> 5a=b+c ------------------ 1
and perimeter = 2010 units
so a+b+c = 2010
but from 1
we get a+5a=2010
so 6a=2010
so a=335
and b+c=5a=1675
now in a right triangle fro Pythagoras theorem we have
c^2 = a^2+b^2
=> c^2-b^2=a^2

=> c^2-b^2=335*335
=> (c-b)(c+b)=335*335
but c+b = 1675
so (c-b)*1675=335*335
so c-b = 67
c+b= 1675
subtracting the above two we get
2b=1608 so b= 804
thus area=1/2 * 335 * 804
therefore area of the triangle=134670 sq. units