Question

rounded to the nearest tenth, what is the perimeter of rectangle abcd?
(image of rectangle abcd with diagonal ac = 5 in, angles at a and c: 60° and 30°)
a. 10.8 inches
b. 13.7 inches
c. 15.0 inches
d. 18.7 inches

Explanation:

Step1: Identify triangle properties

In rectangle \(ABCD\), \(\triangle ACD\) is a right - triangle with hypotenuse \(AC = 5\) inches, \(\angle CAD=60^{\circ}\) and \(\angle ACD = 30^{\circ}\). 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 side (let's call it \(x\)), the side opposite \(60^{\circ}\) is \(x\sqrt{3}\) and the hypotenuse is \(2x\).

Step2: Find the length of \(AD\) and \(CD\)

For \(\triangle ACD\), hypotenuse \(AC = 5\) inches. The side \(AD\) is opposite \(30^{\circ}\) angle. So, if we let \(AD=x\), then \(AC = 2x\). Since \(AC = 5\), we have \(2x=5\), so \(x=\frac{5}{2}=2.5\) inches. So, \(AD = 2.5\) inches.
The side \(CD\) is opposite \(60^{\circ}\) angle. So, \(CD=x\sqrt{3}\), where \(x = 2.5\). Then \(CD=2.5\times\sqrt{3}\approx2.5\times1.732 = 4.33\) inches.

Step3: Calculate the perimeter of the rectangle

The perimeter of a rectangle \(P = 2\times( length+width)\). For rectangle \(ABCD\), length \(AD = 2.5\) inches and width \(CD\approx4.33\) inches.
\(P=2\times(2.5 + 4.33)=2\times6.83 = 13.66\approx13.7\) inches.

Answer:

B. 13.7 inches