We are given that in a right triangle ABC the length of the hypotenuse AB is 12 and that sin x=0.6, and we are asked to evaluate the area of △ ABC. We will name the missing sides a and b.
First let's recall the definition of the sine of an angle.
If△ ABCis a right triangle with acute∠ A,
then the sine of∠ Ais the ratio of the length
of the leg opposite∠ Ato the length of
the hypotenuse.
Using this definition, we can write an equation for sin x.
sin x=a/12
Since we are given that the sine of x is 0.6, we will substitute this value into the equation.
One of the legs of this triangle is 7.2. Next, we can use the Pythagorean Theorem to evaluate the length of the second leg. According to this theorem, the sum of the squared legs is equal to its squared hypotenuse.
Finally, we can evaluate the area of this triangle. Let's recall that the area of a right triangle is half of the product of its legs.
A_(ABC)=1/2( 7.2)( 9.6)=34.56≈ 34.6
The area of △ ABC is approximately 34.6 units^2 and this corresponds with answer D.
