EXERCISES ON BOOK II 31. Two equal triangles are on the same base and on opposite sides of it: shew that the base bisects the straight line joining their vertices. 32. The diagonals of a quadrilateral intersect at right angles: shew that the rectangle contained by the diagonals is double the quadrilateral. 33. The diagonals of a quadrilateral intersect at right angles: shew that the sum of the squares on one pair of opposite sides is equal to the sum of the squares on the other pair. *34. ABC is an equilateral triangle and AD is perpendicular to BC shew that the square on AD is equal to three times the square on BD. 35. Prove that the rectangle contained by the segments of a given straight line is greatest when the segments are equal. 36. If a straight line is divided into two parts, prove that the sum of the squares on the parts is least when the line is bisected. 37. A triangle whose base is one of the non-parallel sides of a trapezium, and vertex the middle point of the opposite side, is equal to half the trapezium. 38. D is the middle point of the base BC of the triangle ABC, BE bisects AD and meets AC at E: shew that the triangle BEC is double the triangle ABE. *39. The difference of the squares on two sides of a triangle is equal to the difference of the squares on the segments of the base made by a perpendicular from the opposite vertex. *40. If O be a point in the base BC, or BC produced, of the isosceles triangle ABC, the difference of the squares on OA and AB is equal to the rectangle contained by OB and OC. 41. If from the middle point of one of the sides of a rightangled triangle a perpendicular be drawn to the hypotenuse, the difference of the squares on the segments of the hypotenuse is equal to the square on the remaining side. 42. Prove that three times the sum of the squares on the sides of a triangle is equal to four times the sum of the squares on the lines drawn from the vertices to the middle points of the opposite sides. 43. The hypotenuses of three isosceles right-angled triangles form a right-angled triangle: shew that one of the isosceles triangles is equal to the sum of the other two. 44. In any quadrilateral the sum of the squares on the diagonals is double the sum of the squares on the lines joining the middle points of the opposite sides. 45. Prove, by constructions similar to that in the First Proof of Theor. 9, that in a triangle the sum of the squares on the sides containing an acute angle is greater, and on those containing an obtuse angle less, than the square on the other side. 46. Construct a square equal to the difference of two given squares. 47. Divide a given straight line into two segments such that the sum of their squares may be equal to the square on another given straight line. When must the straight line be divided externally? 48. Divide a given straight line into segments such that the difference of their squares may be equal to a given square. 49. Bisect a parallelogram by a straight line drawn through a given point. 50. Trisect a triangle by straight lines drawn from a given point in one of its sides. 51. Find a point in a given straight line such that the sum of the squares on its distances from two given points may be the least possible. 52. Find the locus of a point such that the sum of the squares on its distances from two given points may be equal to a given square. 53. Find the locus of a point such that the difference of the squares on its distances from two given points may be equal to a given square. DEFINITIONS OF BOOK II DEF. 1. The altitude of a parallelogram with reference to a given side as base is the perpendicular distance between the base and the opposite side. DEF. 2. The altitude of a triangle with reference to a given side as base is the perpendicular distance between the base and the opposite vertex. DEF. 3. The straight lines drawn through any point in a diagonal of a parallelogram parallel to the sides divide it into four parallelograms, of which the two whose diagonals are upon the given diagonal are called parallelograms about that diagonal and the other two are called the complements of the parallelograms about the diagonal. DEF. 4. All rectangles being identically equal which have two adjoining sides equal to two given straight lines, any such rectangle is spoken of as the rectangle contained by those lines. In like manner, any square whose side is equal to a given straight line is spoken of as the square on that line. DEF. 5. Apoint in a finite straight line is said to divide it internally, or, simply, to divide it; and by analogy, a point in the line produced is said to divide it externally; and, in either case, the distances of the point from the extremities of the line are called the segments of the line. Printed by Cowan & Co. Limited, Perth. |