By Howard Eves
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Extra resources for Mathematical circles adieu: a fourth collection of mathematical stories and anecdotes
If the size of angle ASB = 47◦ , find the size of angle BAS. ANGLE IN THE ALTERNATE SEGMENT 27 C S O 47° A B A B (c) (d) Solution. Write Angle ABS = 90◦ Angle BAS = 43 ◦ Angle in a semicircle is a right angle Angle sum of triangle ABC = 180◦ The theorem that the angle in a semicircle is a right angle has a converse theorem, and is stated here: If there is a right-angled triangle ABC which is right-angled at C (see figure d), and the circumcircle of the triangle is drawn, then line AB is the diameter of the circumcircle.
In the figure, the arc AB of the circle is a minor arc because it is less than a semicircle. References: Line Graph, Networks. ARC LENGTH 37 ARC LENGTH The length of an arc of a circle can be calculated as in the following example. G1 10 A 120° G2 Example. Amanda is schooling up her horse George for a show. He is trotting in a circle at the end of a rope held by Amanda, who is at point A in the figure. The length of the rope is 10 meters. (a) As Amanda turns through an angle of 120◦ find the distance trotted by George along the arc of the circle from G 1 to G 2 .
Example. Figure c shows an upright, square-based pyramid. Calculate the angle between the plane ABCD, which is the base of the pyramid, and the sloping plane VBC. V 5 cm A B O D M C 4 cm (c) Solution. The two lines we select in the planes are VM and OM, as both lines are at right angles to their line of intersection BC. These lines are selected because they are two sides of a right-angled triangle VOM, whose lengths we can find. The required angle between the two planes is angle VMO. 1◦ . References: Angle, Angle between a Line and a Plane, Dihedral Angle, Pyramid, Pythagoras’ Theorem, Square Root, Trigonometry.
Mathematical circles adieu: a fourth collection of mathematical stories and anecdotes by Howard Eves