Thread: Ratio of inscribed and circumscribed triangles

1. Ratio of inscribed and circumscribed triangles

An equilateral triangle is inscribed in a circle, and the circle is inscribed in another equilateral triangle. Find the ratio of the area of the smaller triangle to that of the larger triangle.

Im getting 1:3 but the answer is 1:4.

Thank You

2. Hello, spred!

An equilateral triangle is inscribed in a circle,
and the circle is inscribed in another equilateral triangle.
Find the ratio of the area of the smaller triangle to that of the larger triangle.
Code:
                      A
D ∆ - - - - - - - * ∆ * - - - - - - - ∆ E
\          *    / \    *          /
\       *     /   \     *       /
\     *     /     \     *     /
\         /       \         /
\  *    /         \    *  /
\ *   /     ∆ O   \   * /
\*  /      |      \  */
\ /       |       \ /
C ∆  - - - * - - -  ∆ B
*       D       /
\*           */
\   * * *   /
\         /
\       /
\     /
\   /
\ /
∆
F

The center of the circle is $O.$
The inscribed triangle is $ABC.$
The circumscribed triangle is $D{E}F.$
$O$ is the centroid of $\Delta ABC$ and $\Delta D{E}F.$

Draw segment $OB.$

In right triangle $ODB,\,\text{ let }r = OD.$
. . Then the altitude $AD$ of $\Delta ABC$ is $3r.$

Since $\angle ODB = 30^o\!:\;\;OB = 2r$
. . Then the altitude $DB$ of $\Delta D{E}F$ is $6r$

The ratio of the altitudes is: . $3r: 6r \:=\: 1: 2$

Therefore, the ratio of the area is: . $1^2:2^2 \;=\;1:4$

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equilateral triangle is circumscribed by a circle and another circle is inscribed in that triangle.find the ratio of the areas of two circles

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