Question

select the correct answer.
rounded to the nearest tenth, what is the perimeter of the triangle?
a. 9.3 centimeters
b. 10.8 centimeters
c. 11.0 centimeters
d. 11.8 centimeters
e. 16.2 centimeters

Explanation:

Step1: Find the shorter leg

In a \(30 - 60-90\) triangle, the side opposite the \(30^{\circ}\) angle (shorter leg) is half the hypotenuse.
Let the hypotenuse \(c = 5\) cm. Shorter leg \(a=\frac{c}{2}=\frac{5}{2}=2.5\) cm.

Step2: Find the longer leg

The longer leg \(b\) (opposite \(60^{\circ}\) angle) in a \(30 - 60 - 90\) triangle is \(b = a\sqrt{3}\).
Substitute \(a = 2.5\) cm, \(b=2.5\sqrt{3}\approx2.5\times1.732 = 4.33\) cm.

Step3: Calculate the perimeter

Perimeter \(P=a + b + c\).
Substitute \(a = 2.5\) cm, \(b\approx4.33\) cm, \(c = 5\) cm.
\(P=2.5+4.33 + 5=11.83\approx11.8\) cm.

Answer:

D. 11.8 centimeters