Question

select the correct answer.
rounded to the nearest tenth, what is the perimeter of rectangle abcd?
a. 48.3 cm
b. 54.6 cm
c. 60.0 cm
d. 173.2 cm

Explanation:

Step1: Find the length of \(AB\) and \(AD\)

In right - triangle \(ABC\), \(\cos60^{\circ}=\frac{AB}{AC}\), given \(AC = 20\mathrm{cm}\).
Since \(\cos60^{\circ}=\frac{1}{2}\), then \(AB=\cos60^{\circ}\times AC=\frac{1}{2}\times20 = 10\mathrm{cm}\).
In right - triangle \(ADC\), \(\cos30^{\circ}=\frac{AD}{AC}\), and \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\), so \(AD=\cos30^{\circ}\times AC=\frac{\sqrt{3}}{2}\times20 = 10\sqrt{3}\mathrm{cm}\approx17.3\mathrm{cm}\).

Step2: Calculate the perimeter of the rectangle

The perimeter formula of a rectangle is \(P = 2(l + w)\), where \(l\) is the length and \(w\) is the width. Here \(l = AD\) and \(w = AB\).
\(P=2(AB + AD)=2(10 + 10\sqrt{3})\).
\(P=20+20\sqrt{3}\approx20 + 20\times1.732=20 + 34.64=54.64\approx54.6\mathrm{cm}\).

Answer:

B. \(54.6\mathrm{cm}\)