The longer leg of a right triangle is 1m longer than the shorter leg. T

Question

The longer leg of a right triangle is
1m
longer than the shorter leg. The hypotenuse is
9m
longer than the shorter leg. Find the side lengths of the triangle.

Answer 1

Pythagorean theorem: a squared+ b squared = c  squared or a^2 + b ^2 = c ^2 , c is the hypotenuse,  and the short leg will be x, for the sake of the problem.   a, the short leg is x.  b, the long leg is x+1  and c is x+9.      

      x^2    +   (x+1)^2   =  (x + 9)^2   , to solve (x+1)^2 and (x+9)^2 use FOIL (first, outside, inside , last)

      x^2 + (x+1)(x+1) =  (x+9)(x+9)

    x^2 + (x^2 + 2x + 1) = x^2 +18x + 81

            2x^2 + 2x +1 = x^2 + 18x + 81     , do some subtracting,

           -2x^2  -2x -1 =  - 2x^2 - 2x -1  

                             0  =      -x^2 +16x +  80, switch sides to get

                      x^2 - 16x-80 = 0, the numbers you would subtract to get -16 AND multiply to get -80 is -20 and 4 so,

                     (x-20)(x+4) = 0 , x is either 20 or -4, since -4 is impossible 

                                    x= 20    , Check your work, put 20 as the short leg,  the long leg is 1 meter longer then the short so the long leg is 21, the hypotenuse is 9 meter longer the the short leg so the hypotenuse is 29

a (short leg) = 20     b (long leg) = 21     c(hypotenuse) =29 , a^2 + b ^2 = c^2 

                 20^2 + 21^2 = 29 ^2  , 

                 400   + 441 = 841, which is true so  the short leg is 20, the long leg is 21 and the hypotenuse is 29

                    

                                  

Answer 2

Let a be the short leg the right triangle, b be the longer leg, and c be the hypotenuse.

• The longer leg of a right triangle is 1m longer than the shorter leg: b = a+1
• The hypotenuse is 9m longer than the shorter leg: c = a + 9

 

Use the Pythagorean Theorem:

 

a² + b² = c²

a² + (a+1)² = (a+9)²

a² + a² + 2a + 1 = a² + 18a + 81

a² - 16a - 80 = 0

 

Factors to:

 

(a-20)(a+4) = 0

a = 20 and -4

 

Since we can't have a length of -4, a = 20, b = a+1 = 21, c = a+9 = 29

 

Check:

20² + 21² = 29²

400 + 441 = 841

841 = 841

Check!