hypotenuse question

Question

a lot has the shape of a right triangle. the difference between the two sides is 2 meters. the hypotenuse is 6 meters less than twice the length of the longer side. find the length of the shorter side

Answer 1

Here's what we know:

a² + b² = c2 (Pythagorean theorem)

a - b = 2 (the difference of the two sides is 2)

c = 2a - 6 (the hypotenuse is 6 less than 2twice the long side)

 

We can use substitution to isolate down to one variable:

-b = 2 - a

b = a - 2

 

a² + (a - 2)² = (2a - 6)²

FOIL:

a² + a2 - 2a - 2a + 4 = 4a² - 12a - 12a + 36

Combine like terms:

2a² - 4a + 4 = 4a² - 24a + 36

Move everything to one side:

0 = 2a² - 20a + 32

Factor out a 2:

0 = 2(a² - 10a + 16)

Divide both sides by 2:

0 = a² - 10a + 16

Now factor out the quadratic:

0 = (a - 8)(a - 2)

a = 8 or 2

Since b = a - 2, a cannot be 2, else b = 0

a = 8

b = 8 - 2

b = 6

 

check:

8² + 6² = c²

64 + 36 = c²

100 = c²

10 = c

 

10 = 2 * 8 - 6

10 = 16 - 6

10 = 10