# The A to Z of mathematics a basic guide - Sidebotham T.H.

ISBN 0-471-15045-2

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♦ One line parallel to another

1. Construction of an angle of 60°. Draw a straight line with a point A on it (see

figure a). Open the compasses to any radius, and with the center at A draw a large arc, and where this arc cuts the straight line call it point B. Using the same radius, with the center at B, draw an arc to cut the first arc at point C. Join points A and C with a straight line. Angle BAC is 60°.

CONSTRUCTIONS 111

(a)

2. Construct an angle of 30°. First construct an angle of 60°. Then bisect the 60° angle to make an angle of 30°. See the entry Angle Bisector for this method.

3. Construct an angle of 90°. Draw a straight line with a point A on the line (see figure b). Open compasses to any radius, and with the center at A draw a semicircle to cut the line at the points C and D. Open the compasses out a little further, and with the center at point C draw an arc. Using the same radius, and with the center at D, draw another arc to cut at the point E that arc you have just drawn. Draw the line AE. Angle DAE is an angle of 90°, and AE is the perpendicular bisector of the line segment CD.

(b)

4. Construct an angle of 45°. First construct an angle of 90°. Then bisect the 90° angle to make an angle of 45°. See the entry Angle Bisector for this method.

5. Construct a regular hexagon, which has six sides each equal to a length of 5 cm. First mark a point O where you want the center of the regular hexagon to be. (See figure c). With the center at 0, draw a circle of radius 5 cm. Mark a point A at the top of the circle, or at any point you wish, on the perimeter of the circle. Using the same radius of 5 cm, starting at the point A, step off the other vertices

112 CONSTRUCTIONS

B, C, D, E, and F. Join up the points A, B, C, D, E, and A to make the regular hexagon.

A

6. Construct a rectangle with sides of lengths 6 and 4 centimeters. Draw a line with a point A on it (see figure d ). From A measure a distance of 6 centimeters to another point B. At point A and at point B construct two lines at 90° to the line AB, using the method described earlier. Along the first line measure a distance of 4 centimeters and mark the point D, and on the second line do likewise and mark the point C. Join the points C and D with a straight line. The required rectangle is ABCD.

D C

y r'”

r./'- ”''V S?'~~

/ 1 i U .

A B

(d)

7. Construct one line parallel to another. In figure e, AB is a straight line and P is a point that is not on the line. Construct a straight line that is parallel to the line AB, and also passes through the point P. Center the compasses on the point A and with a radius of length AP draw an arc to intersect the line AS at a point called C. Using the same radius, with the center at the point C draw an arc that will intersect at Q a similar arc drawn from the point P. Draw the line PQ. The line PQ is parallel to the line AB.

CONVERSE OF A THEOREM 113

• 1 • ----------------------------------

A 'C B

(e)

References: Angle, Arc, Angle Bisector, Bisect, Hexagon, Perpendicular Bisector, Parallel, Radius, Rectangle.

CONTINUOUS CURVE

Reference: Discontinuous.

CONTINUOUS DATA

Reference: Data.

CONTINUOUS DISTRIBUTION

Reference: Frequency Distribution.

CONVERSE OF A THEOREM

This applies to theorems in geometry, but also it can apply to other mathematical statements or assertions. Refer to the entry Cyclic Quadrilateral for an example. The converse of a theorem is best explained by quoting a well-known theorem and then its converse. The example quoted here is Pythagoras’ theorem, which can be written as an “If... then...” statement (see figure a):

A

114 CONVERSE OF A THEOREM

If a triangle ABC is right-angled at B, then b2 = a2 + c2.

The converse of Pythagoras’ theorem reverses the “If... then...” statement:

Ifb2 = a2 +c2 is true for a triangle, then the triangle is right-angled at B.

In Pythagoras’ theorem, the theorem and its converse are both true, but this is not the case with every theorem. When the ancient Egyptians built the square-based pyramids, they were able to construct very accurate right angles using the converse of Pythagoras’ theorem. The Great Pyramid of Gizeh was erected about 2900 BC and it has been calculated that the right angles of its base are accurate to within 1 part in 27,000. It is not known for certain how the builders achieved that degree of accuracy, but it is thought that the following method might have been used.

A rope in the shape of a loop was divided up into 12 equal parts by 12 knots (see figure b). The rope was arranged into the shape of a triangle with sides of length 3,4, and 5 units. Three stakes were fixed into the ground, one at each vertex of the triangle, to ensure the rope was taut. The angle in the triangle that was opposite to the longest side was the right angle.

Greater accuracy is obtained using larger triangles. Comers of a soccer pitch can be marked out using three people and three measuring tapes of lengths 300,400, and 500 meters.

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