Algebra Word story problem

Question

one leg of a right triangle is 8 millimeters shorter than the longer leg and the hypotenuse is 8 millimeters longer than the longer leg. Find the lengths of the side of the traingle

Answer 1

Some Word (Story) problems have TMI (too much information) or information that we use in a different form.  This problem uses the hypotenuse (always the longest side) of a triangle.

 

Let A,B,C be the leg lengths (C is the hypotenuse because the Pythagorean theorem is written that way).

 

Translate:

 "one leg of a right triangle is 8 millimeters shorter than the longer leg" means

         A                             =                -8               +              B 

 "the hypotenuse is 8 millimeters longer than the longer leg"  means

               C          =   8                 +                       B

 "Find the lengths of the side of the triangle"   means   report A, B, C

 

The Math:

     C²    =    A²    +    B²                    [Pythagorean Theorem]

   (B+8)²  =  (B-8)²  +   B²              [substitute for C and A using equations above]

  B² + 16B  + 64  =  B²  -16B  + 64    + B²                [expand]

          16B  = B²  - 16B                 [subtract B²+64 from both sides]

          32B  = B²                   [add 16B to both sides]

           32   =  B                   [divide both sides by B]

 

Also,     A = B-8 = 24

            C = B+8 = 40

 

      A=24,  B=32,  C=40

 

Checking (very important):

    Is  24, 32, 40 a right triangle   ?

         40²  =  24² + 32²

        1600 =  576  +   1024   ?

            1600  =  1600   ? yes

 

Also note that  3n-4n-5n is what's called a Pythagorean Triple, so triangles like 3-4-5, 6-8-10, 9-12-15, ... 24-32-40 are right triangles.

Answer 2

Hi Joe, let x be the length of the longer leg. Then x - 8 is the length of the shorter leg and x + 8 is the length of the hypotenuse.

 

Use the Pythagorean Theorem to solve for x.

 

x² + (x - 8)² = (x + 8)²

 

x² + x² - 16x + 64 = x² + 16x + 64

 

Combine terms:

 

x² - 32x = 0

 

Complete the square:

 

(1 / 2 * -32)² = 16²

 

Add 16² to both sides:

 

x² - 32x + 16² = 16²

 

Factor the left side:

 

(x - 16)² = 16²

 

Take the square root of both sides:

 

x - 16 = 16

 

Add 16 to both sides:

 

x = 32

 

The longer leg is 32 millimeters in length.

 

The shorter leg is: 32 - 8, or 24 millimeters in length.

 

The hypothenuse is: 32 + 8, or 40 millimeters in length.

 

Questions?