Answer:
The area of the equilateral triangle is 216 sq. inches.
Step-by-step explanation:
Given : A circle has a radius of 6 in.
To find : The circumscribed equilateral triangle will have an area of?
Solution :
We draw a rough sketch of the given situation,
Form an equilateral triangle PQR in which a circle in circumscribed with radius r and center O.
Refer the attached figure below.
Radius of the circle is 6 inches.
In ΔPQR,
Top find the area of the equilateral triangle we need to find the length of the base(b) i.e. QR and it's height(h) i.e. PT.
Area of the triangle is [tex]A=\frac{1}{2}\times b\times h[/tex]
Now we apply the trigonometric identity in ΔOTR.
Since, QR=2 TR
In ΔOTR,
[tex]\sin 30=\dfrac{OT}{TR}\\\\\dfrac{1}{2}=\dfrac{6}{TR}\\\\TR=12[/tex]
QR=2 TR=2(12)=24 in is base of the triangle.
Now in ΔPOS,
[tex]\cos 60=\dfrac{OS}{PO}\\\\\dfrac{1}{2}=\dfrac{6}{PO}\\\\PO=12 in.[/tex]
AD=AO+OD
AD=12+6=18 in. is the height of the triangle.
Hence,
Area of the triangle is
[tex]A=\frac{1}{2}\times b\times h[/tex]
[tex]A=\frac{1}{2}\times 18\times 24[/tex]
[tex]A=216[/tex]
Therefore, The area of the equilateral triangle is 216 sq. inches.
