Question

if \\(sqrt{61}\\) is the longest side length in the triangle, find the value of ( x ) that makes the triangle above a right triangle. write your answer in simplest radical form.
a. ( 5 )
b. ( 6 )
c. ( 2sqrt{15} )
d. ( 6sqrt{10} )

Explanation:

Step1: Apply Pythagorean theorem

$$x^2 + (x+1)^2 = (\sqrt{61})^2$$

Step2: Expand and simplify equation

$$x^2 + x^2 + 2x + 1 = 61$$
$$2x^2 + 2x - 60 = 0$$
$$x^2 + x - 30 = 0$$

Step3: Factor quadratic equation

$$(x+6)(x-5) = 0$$

Step4: Solve for positive x

Since length can't be negative, $x=5$

Answer:

A. 5