28. Two lots, one circular and the other square, are each 120 rods in perimeter; which has the greater area, and how much? 31. The area of a regular hexagon inscribed in a circle is twice the area of an equilateral triangle inscribed in the same circle. 32. The side of a square circumscribed about a circle is equal to the diagonal of a square inscribed in the same circle. 33. If the points of bisection of the sides of a given triangle be joined, the triangle so formed will be one-fourth of the given triangle. 34. To divide a triangle into two parts by a perpendicular to the base, so that these parts shall be proportional to two given lines. 35. By a line parallel to a side, to divide a triangle in the ratio of two given lines. 36. If two points be taken in the diameter of a circle, equally distant from the centre, the sum of the squares of two lines drawn from these points to any point in the circumference, is constant. 37. To divide a circle into n equal parts by concentric circles. Suggestion. On the radius of given circle describe a semicircle. Divide given radius into the required number of equal parts. (Ch. IX, Prob. III.) At the points of division, erect perpendiculars, cutting the semicircumference. Join points of division with given centre. The proof is based on the similarity of triangles and Th. VII. 38. On a given base, to construct a regular hexagon. B D CHAPTER XI. MAXIMA AND MINIMA. DEFINITIONS. 1. As previously defined, the Perimeter of a polygon is the sum of its sides. 2. Two polygons are Isoperimetric when their perimeters are equal; that is, when the distance around each is the same. 3. A Maximum quantity is the greatest of its kind. 4. A Minimum quantity is the least of its kind. Axiom.-The minimum distance between two points is a straight line. THEOREM I. The minimum distance from a point to a line is a straight line perpendicular to the given line (See Ch. II, Th. XIX, Cor.). THEOREM II. The maximum line which can be inscribed in a circle, is a diameter (See Ch. IV, Th. XII, Exs. 1 and 7; also, Ch. II, Th. III). THEOREM III. The minimum line which can be inscribed in a circle through a given point, is the chord perpendicular to the line joining the given point and the centre (See Ch. IV, Th. XII, Ex. 2). THEOREM IV. The sum of the lines drawn from two fixed points on the same side of a line to a point in that line, is a minimum when the lines make equal angles with the given line. F For, let P' be any other point in A B. Let fall the perpendicular C E, and produce it till E F = CE. Join P' with F, C, and D. Among triangles formed with the same two given sides, that in which these two sides are perpendicular to each other, is the these two sides are perpendicular to each other, be greater than A B E, in which they are oblique to each other. .'. A C D E, since A C = A E, by supposition. ABX AC>ABX DE. .. ABC ABE. THEOREM VI. Q. E. D. Among triangles having equal bases and equal areas, the isosceles triangle has the minimum perimeter. Let A B C be an isosceles triangle; and let A B E be any other triangle of equal base and equal area, but not isosceles; then will A B A C + B C <ABAE + BE. For, the base and area being constant, the vertex will always be in the A4 same line, CD, parallel to the base. H Cid E -D a F B Draw B F perpendicular to CD, and produce it to H, making FHBF. Join C and H, E and H. Then, and But, But, b+c+ d = a + b + c = 2 R. AH is a straight line. In the triangle A EH, A H < A E + E H. .. AB+ AC+ BC < A B+ AE + BE. Q. E. D. EXERCISE. Among parallelograms having equal bases and equal areas, which has the minimum perimeter? THEOREM VII. The sum of two adjacent sides of a rectangle being constant, the area is a maximum when the sides are equal. E G H α D B Let A B represent the constant sum; AD and D B representing the two adjacent sides when they are equal, and AC and BC representing them when they are unequal; then will ADX BD > ACX BC. For, on A B as a diameter, describe a circle. Through C and D, draw the perpendiculars E F and G H. |