Question

What is the value of x in the triangle?a right triangle with a short leg of length x and hypotenuse of length 3 times the square root of 2A. B. C. D. E.

Answer 1

Solution:

Given the right triangle below:

To solve for x, we use the trigonometric ratio.

In the above triangle, the angles at A and B are equal.

Thus, we have

$\begin{gathered} \angle A+\angle B+\angle C=180(su\text{m of angles in a triangle\rparen} \\ \angle A=\angle B \\ thus, \\ 2\angle\text{B+90=180} \\ \Rightarrow2\angle\text{B=180-90} \\ 2\angle\text{B=90} \\ \Rightarrow\angle\text{B=45} \end{gathered}$

From trigonometric ratio,

$\sin\theta\text{=}\frac{opposite}{hypotenuse}$

In this case, θ is the angle at B, which is 45; opposite is AC, and hypotenuse is AB.

Thus,

$\begin{gathered} \sin45=\frac{x}{3\sqrt{2}} \\ \Rightarrow x=3\sqrt{2}\times\sin45 \\ =3\sqrt{2}\times\frac{1}{\sqrt{2}} \\ =3 \end{gathered}$

Hence, the value of x is

$3$

The correct option is B