Geometry: An Interactive Journey to Mastery

Lesson 22: Explorations with Pi

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A small circle is inscribed in an equilateral triangle, which is inscribed in a larger

circle of radius 2.

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The small triangle shown in )LJXUH is half an

equilateral triangle. Thus, a = 1 and b  2122 3.

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2 ˜˜ 233 33.

The area of the shaded region we seek is 33 .S

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x Many geometry curricula have students memorize the side lengths of right

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If the short side of the triangle has length q, then the hypotenuse is double

this, 2q, and the Pythagorean theorem then gives the length of the third side

as 3.q Because many textbook problems are designed to make use of

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triangles in a margin and compute their side lengths.

Figure 22.4

2

a^2

b

a^2

b30°

Figure 22.5

2 q 30°

q

3 q

Figure 22.6