Question

a napkin is folded into an isosceles triangle, triangle abc, and placed on a plate, as shown. the napkin has a perimeter of 38 centimeters. to the nearest square centimeter, how many square centimeters of the plate are covered by the napkin? trigonometric area formula: area = $\frac{1}{2}absin(c)$ 16 square centimeters 30 square centimeters 56 square centimeters 60 square centimeters

Explanation:

Step1: Find the lengths of the equal - sides

Let the equal - sides of the isosceles triangle be \(a\) and the base be \(b = 8\) cm. The perimeter \(P=a + a + b=2a + b\). Given \(P = 38\) cm and \(b = 8\) cm, we have \(2a+8 = 38\).
\[2a=38 - 8=30\]
\[a = 15\] cm.

Step2: Calculate the area of the triangle

We use the trigonometric area formula \(A=\frac{1}{2}ab\sin(C)\). Here, \(a = 15\) cm, \(b = 8\) cm, and \(C = 30^{\circ}\), and \(\sin(30^{\circ})=\frac{1}{2}\).
\[A=\frac{1}{2}\times15\times8\times\sin(30^{\circ})\]
\[A=\frac{1}{2}\times15\times8\times\frac{1}{2}\]
\[A = 30\] square centimeters.

Answer:

30 square centimeters