Question

select the correct answer. what is the value of x in the triangle? image of right triangle with hypotenuse ( 3sqrt{2} ), right angle, two other angles equal
a. ( 3sqrt{2} )
b. ( 3 )
c. ( 6 )
d. ( 6sqrt{2} )
e. ( 2sqrt{2} )

Explanation:

Step1: Identify Triangle Type

The triangle is a right - isosceles triangle (two acute angles are equal, so the two legs are equal, and hypotenuse is given as \(3\sqrt{2}\)). For a right - isosceles triangle, if the legs have length \(x\) and hypotenuse \(h\), the Pythagorean theorem is \(x^{2}+x^{2}=h^{2}\), which simplifies to \(2x^{2}=h^{2}\).

Step2: Substitute Hypotenuse Value

We know \(h = 3\sqrt{2}\), so substitute into the formula \(2x^{2}=(3\sqrt{2})^{2}\). First, calculate \((3\sqrt{2})^{2}=3^{2}\times(\sqrt{2})^{2}=9\times2 = 18\). Then our equation becomes \(2x^{2}=18\).

Step3: Solve for \(x\)

Divide both sides of the equation \(2x^{2}=18\) by 2: \(x^{2}=\frac{18}{2}=9\). Then take the square root of both sides. Since \(x\) represents the length of a side of a triangle, we take the positive square root, so \(x = \sqrt{9}=3\).

Answer:

B. 3