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Largely considered as a vintage of contemporary arithmetic, this multiplied model of Felix Klein's celebrated 1894 lectures makes use of modern ideas to check 3 recognized difficulties of antiquity: doubling the amount of a dice, trisecting an perspective, and squaring a circle. brand new scholars will locate this quantity of specific curiosity in its solutions to such questions as: less than what conditions is a geometrical development attainable?

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What are the radii of the three circles? Solution: Without a picture given, we must draw one. We stop to draw three circles that R are externally tangent to each other. The two largest circles contain the triangle A side of length 15. In the drawing we T have, RI = 9, MR = 11, and MI = 15. Let’s give RA the value of x and see what we can do. If RA is x, then MA = 11–x I since MR = 11. All radii in a circle are N M congruent so MN = 11–x also. Since MI = 15, we know that NI = 15 – (11–x) = 4 + x.

The interior angles of any triangle must add to 180° so m∠POI = 140°. Thus, the measure of arc PI is 140°. O I Example: Find the radius of a circle in which a chord of length 8 is a distance 3 from the center. Solution: The distance to a chord from the center is given to be the perpendicular distance. We then have the drawing here; the radius of the circle is the hypotenuse of a right triangle. It is a recognizable 3, 4, 5 right triangle so the radius of the circle is 5. 3 8 4 Example: Three circles are externally tangent to each other.

O I Example: Find the radius of a circle in which a chord of length 8 is a distance 3 from the center. Solution: The distance to a chord from the center is given to be the perpendicular distance. We then have the drawing here; the radius of the circle is the hypotenuse of a right triangle. It is a recognizable 3, 4, 5 right triangle so the radius of the circle is 5. 3 8 4 Example: Three circles are externally tangent to each other. A triangle joining the centers of the three circles has sides 9, 11, and 15.