Question

find the expression that represents the length of the hypotenuse of a right triangle whose legs measure ( m^2 - n^2 ) and ( 2mn ).
a. ( sqrt{m^4 + 6m^2n^2 + n^4} )
b. ( sqrt{m^4 + n^4} )
c. ( m^4 + 2m^2n^2 + n^4 )
d. ( m^2 + n^2 )
please select the best answer from the choices provided
a
b
c
d

Explanation:

Step1: Apply Pythagorean theorem

Let hypotenuse = $c$, legs $a=m^2-n^2$, $b=2mn$.
$$c^2 = (m^2-n^2)^2 + (2mn)^2$$

Step2: Expand each squared term

$$(m^2-n^2)^2 = m^4 - 2m^2n^2 + n^4$$
$$(2mn)^2 = 4m^2n^2$$

Step3: Sum the expanded terms

$$c^2 = m^4 - 2m^2n^2 + n^4 + 4m^2n^2 = m^4 + 2m^2n^2 + n^4$$

Step4: Factor the perfect square

$$c^2 = (m^2 + n^2)^2$$

Step5: Take square root

$$c = m^2 + n^2$$

Answer:

D. $m^2 + n^2$