Equilateral Triangle

magentarita · #1 $Old$ November 20th, 2008, 07:53 AM

Find the perimeter of the equilateral triangle inscribed in a circle of radius 20.0 inches

Soroban · #2 $Old$ November 20th, 2008, 08:31 AM

Hello, magentarita!

Quote:

Find the perimeter of the equilateral triangle inscribed in a circle of radius 20 inches

Code:

                A
              * * *
          *    /|\    *
        *     / | \     *
       *     /  |  \     *
            / 20|   \  D
      *    /    |    *    *
      *   /     *     \   *
      *  /      O      \  *
        /               \
     B *- - - - - - - - -* C
        *               *
          *           *
              * * *

The triangle is $ABC$
The circle has center $O$ and radius $OA = 20$
Draw $OD \perp AC.$

Since $\Delta ABC$ is equilateral, $\angle A \:=\:\angle B \:=\:\angle C \:=\:60^o$
. . Then: $\angle OAD = 30^o$

In right triangle $ADO\!:\;\;\cos30^o \:=\:\frac{AD}{20} \quad\Rightarrow\quad AD \:=\:20\cos30^o \:\approx\:17.32$ inches

The side of the triangle is: . $2\times 17.32 \:=\:34.64$ inches

Therefore, the perimeter is: . $3 \times 34.64 \:=\:103.92$ inches.

magentarita · #3 $Old$ November 20th, 2008, 04:13 PM

Quote:

Originally Posted by Soroban $View Post$

Hello, magentarita!

Code:

                A
              * * *
          *    /|\    *
        *     / | \     *
       *     /  |  \     *
            / 20|   \  D
      *    /    |    *    *
      *   /     *     \   *
      *  /      O      \  *
        /               \
     B *- - - - - - - - -* C
        *               *
          *           *
              * * *

The triangle is $ABC$
The circle has center $O$ and radius $OA = 20$
Draw $OD \perp AC.$

Since $\Delta ABC$ is equilateral, $\angle A \:=\:\angle B \:=\:\angle C \:=\:60^o$
. . Then: $\angle OAD = 30^o$

In right triangle $ADO\!:\;\;\cos30^o \:=\:\frac{AD}{20} \quad\Rightarrow\quad AD \:=\:20\cos30^o \:\approx\:17.32$ inches

The side of the triangle is: . $2\times 17.32 \:=\:34.64$ inches

Therefore, the perimeter is: . $3 \times 34.64 \:=\:103.92$ inches.

It's amazing how you used trig to answer a geometry question.