Question

Find the area of a 30-60-90 triangle with a hypotenuse that measures 17 cm.

Answer 1

First, let's start by constructing the triangle (not scalable), like this:

Where b and h are the base and the height of the figure.

In order to determine the value of b and h we can use the following trigonometric function, the sine:

$\sin (\theta)=\frac{a}{17}$

Where 17 is the length of the hypotenuse and a is the length of the side opposite to θ. From this formula, we can solve for a by multiplying by 17 on both sides to get:

$a=17\sin (\theta)$

As you can see, in the figure, the side opposite to the 60° angle is b, then by replacing 60 for θ and b for we get:

$b=17\sin (60)$

Then the value of b is calculated to get:

$b\approx14.72$

Similarly, we can get the value of h by replacing 30° for θ and h for a, like this:

$h=17\sin (30)=8.5$

Now we can use the following formula to calculate the area of the triangle:

$A=\frac{b\times h}{2}$

By replacing 8.5 for h and 14.72 for b we get:

$A=\frac{14.72\times8.5}{2}=62.57$

Then, the area of this triangle equals 62.57 square centimeters