Question

Does (c/a - a/c) ÷ b/c = b/a?

Please include all steps in your answer and state if it is True or False.

Answer 1

Answer: True - if Pythagorean Theorem can be used. Otherwise, the answer is False.

Explanation:

`((c)/(a) - (a)/(c) )` ÷
`(b)/(c)` =
`(b)/(a)`

→
`((c)/(a)((c)/(c)) - (a)/(c)((a)/(a)))`÷
`(b)/(c)`

→
`((c^(2))/(ac) - (a^(2))/(ac))` ÷
`(b)/(c)`

→
`(c^(2)-a^(2))/(ac)` ÷
`(b)/(c)`

→
`(c^(2)-a^(2))/(ac)` *
`(c)/(b)`

→
`(c^(2)-a^(2))/(ab)`

***If a, b, c represents the sides of a triangle, then the Pythagorean Theorem can be used: a² + b² = c² ⇒ b² = c² - a²

→
`(b^(2))/(ab)`

→
`(b)/(a)`

Answer 2

False

Explanation:

Multiply both sides by b ÷ c

I know this isn't usual, but there is nothing illegal about it.

(c/a - a/c) = b÷c * b/a Change b÷c to b/c

c/a - a/c = b/c * b/a Combine the right

c/a - a/c = b^2/(ac) Find the numerator using the LCM ac

(c*c - a*a)/(ac) = b^2/(ac) Multiply both sides by ac

c^2 - a^2 = b^2 This may or may not be true. If there is one false case it is not true.

The statement is false.

c = 10

a = 8

b = 6

10^2 - 8^2 = 6^2 This works.

12^2 - 11^2 = 9^2 will not work.