Question

select the correct answer.
rounded to the nearest tenth, what is the perimeter of rectangle abcd?
image of rectangle abcd with diagonal ac = 20 cm, angles at a and c labeled 60° and 30°
a. 48.3 cm
b. 54.6 cm
c. 60.0 cm
d. 173.2 cm

Explanation:

Step1: Identify triangle properties

In rectangle \(ABCD\), \(\triangle ACD\) (or \(\triangle ABC\)) is a 30 - 60 - 90 triangle with hypotenuse \(AC = 20\) cm. In a 30 - 60 - 90 triangle, the sides are in the ratio \(1:\sqrt{3}:2\), where the side opposite \(30^{\circ}\) is the shortest, let's call it \(x\), the side opposite \(60^{\circ}\) is \(x\sqrt{3}\), and the hypotenuse is \(2x\).

For \(\triangle ACD\), angle at \(A\) is \(60^{\circ}\), angle at \(C\) is \(30^{\circ}\), so side \(AD\) (opposite \(30^{\circ}\)): Let \(AD=x\), hypotenuse \(AC = 20\) cm. Since hypotenuse \(= 2x\), then \(2x=20\), so \(x = 10\) cm. Side \(CD\) (opposite \(60^{\circ}\)) is \(x\sqrt{3}=10\sqrt{3}\approx10\times1.732 = 17.32\) cm.

Step2: Calculate perimeter of rectangle

The perimeter of a rectangle \(P=2\times(\text{length}+\text{width})\). Here, length \(CD\approx17.32\) cm and width \(AD = 10\) cm. So \(P = 2\times(10 + 17.32)=2\times27.32 = 54.64\approx54.6\) cm (rounded to the nearest tenth).

Answer:

B. 54.6 cm