SAT practice problem?

Question

A right triangle has side lengths of x-1, x+1, and x+3. What is its perimeter?

Answer 1

Those explanations are way too complicated.  The student should realize that most likely he's working with a 3x4x5x triangle since the answer needs to be an integer.  I'd start with 3,4,5 and see if that worked then move on to 6,8,10.  No need to spend tons of time doing the Algebra, that's exactly what the ETS wants test takers to do.  

Answer 2

The longest side of a right triangle is the hypotenuse.

 

(x - 1)² + (x + 1)² = (x + 3)²

x² - 2x + 1 + x² + 2x + 1 = x² + 6x + 9

2x² + 2 = x² + 6x + 9

x² - 6x - 7 = 0

(x - 7)(x + 1) = 0

 

{-1, 7}

 

Measurements are only positive so 7 is the only credible solution.

 

The sides are 6, 8, and 10. The perimeter is 24.

 

Should you encounter a question of this type on the test, save it for the very end. It is expensive as far as time is concerned.

 

Answer 3

The perimeter is the sum of the lengths of the sides:

 

Perimeter = (x+1) + (x-1) + (x+3) = 3(x+1)

 

But since it's a right triangle, the Pythagorean theorem also holds:

 

(x+1)² + (x-1)² = (x+3)²

 

When you FOIL that out and combine terms, you get:

 

x² - 6x - 7 = 0

 

Which factors to:

 

(x-7)(x+1) = 0

x = 7 and -1

 

We can't have a negative length, so x = 7 and the perimeter is:

 

Perimeter = 3(x+1) = 3(7+1) = 3*8 = 24