La topologie des espaces metriques by Elisabeth Burroni

By Elisabeth Burroni

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Famous problems of elementary geometry: the duplication of the cube, the trisection of an angle, the quadrature of the circle: an authorized translation of F. Klein's Vorträge

Generally considered as a vintage of recent arithmetic, this elevated model of Felix Klein's celebrated 1894 lectures makes use of modern strategies to ascertain 3 well-known difficulties of antiquity: doubling the amount of a dice, trisecting an attitude, and squaring a circle. trendy 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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A 92° B 102° C 88° D 44° 4. 5. Ϫ3 QT If the slope of WY is , which represents 4 Q T T Q the slope of XZ that makes XZ parallel to QT WY? Ϫ3 F 4 Ϫ4 G 3 3 H 4 J 4 3 Find the value of x that makes line m parallel to line n. 7. 105Њ n t 80Њ A B C D 8. 4(x – 10) p 100Њ n m Which pair of lines are parallel? pʈn nʈt nʈp pʈt Find the value of x that makes j ʈ p. x j 4(x – 13) A B C D 6. 33 38 44 p 58Њ Which coordinates for F will make FG ʈ HJ ? 5 J 58 9. Use only the marked angles. H (6, 3) a b G ( –2, 0) x J (2, –3) 120Њ t 120Њ F G H J (0, 3) (3, 0) (3, Ϫ1) (1, 2) Why is line a parallel to line b?

X A Y Z Use your straightedge to draw a line through A and Z. ᐉ X A Y Q T Result: ZA is perpendicular to line ᐉ and passes through the point A. Patty Paper: Step 1. Copy the given sketch on a piece of patty paper. A ഞ Constructions with Compass and Straightedge—Working with Angles • 27 Step 2. Fold the line ᐉ on top of itself until the folded edge passes through the point A. A ഞ Step 3. Unfold and flatten. ഞ A Result: The crease is the line perpendicular to line ᐉ passing through the point A.

Label the intersection Z. ᐉ X A Y Z Use your straightedge to draw a line through A and Z. ᐉ X A Y Q T Result: ZA is perpendicular to line ᐉ and passes through the point A. Patty Paper: Step 1. Copy the given sketch on a piece of patty paper. A ഞ Constructions with Compass and Straightedge—Working with Angles • 27 Step 2. Fold the line ᐉ on top of itself until the folded edge passes through the point A. A ഞ Step 3. Unfold and flatten. ഞ A Result: The crease is the line perpendicular to line ᐉ passing through the point A.

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