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

Question

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

Answer 1

Let the sides be a,b,c and a be the shorter leg and b be the longer leg of the right triangle.

 

Shorter leg is 9 ft shorter than the longer leg. So,

 

a = b - 9 ----- equation (1)

 

The hypotenuse is 9 ft longer than the longer leg. So,

 

c = b + 9 ----- equation (2)

 

Substitute a and c expressed interns of b in to the pythagorean equation.

 

a² + b² = c²

 

(b-9)² + b² = (b+9)²  

 

b² - 18b + 81 + b² = b² + 18b + 81 -------- expand the squared binomials

 

b² - 18b + 81 + b² = b² + 18b + 81 -------- Cancel 81 and b² from both sides

 

b² - 18b = 18b

 

b² - 18b - 18b = 18 b - 18b --------------- subtract 18 b from both sides

 

b² - 36b = 0 

 

b (b-36) = 0 --------- factor b

 

b = 0 or (b-36) = 0

 

b = 36.

 

Substitute b = 36 to find a and c

 

a = b - 9 --------- equation (1)

 

a = 36 - 9

 

a = 27.

 

c = b + 9 ----- equation (2)

 

c = 36 + 9

 

c = 45.

 

Sides of the triangle are a=27 ft, b=36 ft and c=45 ft.

 

Answer 2

We've got a triangle with sides a, b, and c, and I'll say they're in ascending order of length. So the constraints are

a + 9 = b
b + 9 = c or a + 18 = c

a² + (a + 9)² = (a + 18)²

2a² + 18a + 81 = a² + 36a + 324

a² - 18a - 243 = 0

One value of a is negative. Ignore that and get

a = 27
b = 36
c = 45

Sanity check: pop your values for a and b into the Pythagorean Theorem and make sure you get the right c. Cheers! (

 

And oh yeah: no need to ask the same question twice. LOL