The above methods and illustrations will be useful to the student in directing him how to proceed in the investigation of an exercise, but no method can be given which will necessarily succeed in all cases. In such a branch of study only careful observation of the methods employed in different cases and extensive practice in their application can give confidence and ensure success. EXERCISES. BOOK I. Sides and Angles of Triangles. Conditions of Equality. Props. 4, 8, 26, 5 and 6. 1. The perpendiculars from the middle points of two sides of a triangle meet in a point which is equidistant from the three angles of the triangle. 2. Find a point within an acute-angled triangle from which a circle may be described about the triangle. 3. Two angles of an equilateral triangle are bisected, and the point at which the bisecting lines intersect is joined with the third angle; show that the third angle will be bisected. 4. The exterior angles at the base of an isosceles triangle ABC are bisected by the straight lines AO, BO intersecting in 0. If AO be joined show that it will bisect the vertical angle ACB. 5. In any isosceles triangle if a straight line be drawn from the vertex to the base and fulfil any one of the following conditions it must fulfil the other two. The conditions are that the line 1, bisects the base, 2 bisects the vertical angle, 3 is perpendicular to the base, (Three exercises.) [These properties of an isosceles triangle form the key to the next three exercises.] 6. To find a point' in a given straight line which shall be equidistant from two given points. 7. Through a given point to draw a straight line equally inclined to two given lines. 8. From two given points on opposite sides of a line draw two straight lines which shall meet in that given straight line and include an angle bisected by the given straight line. 9. The diagonals of a rhombus bisect each other at right angles. io. Inscribe a rhombus in a given triangle, having one of its angles coincident with an angle of the triangle. 11. Show that the perpendiculars dropped from any point in the line bisecting an angle of a triangle upon the sides containing that angle, or those sides produced, will be equal to one another. a 12. Find a point in a given straight line from which the perpendiculars upon two given straight lines which intersect shall be equal to one another. Sides and Angles of Triangles Conditions of Inequality. Props. 18, 19, 20, 21, 24, 25. 13. If AD is the longest side and BC the shortest of a quadrilateral ABCD, then the angle ABC is greater than ADC, and the angle BCD greater than BAD. 14. In any quadrilateral figure show that any one side is less than the sum of the other three. 15. The four sides of a quadrilateral are together greater than the sum and less than twice the sum of the diagonals. 16. The two sides of a triangle are together greater than twice the line joining the vertex with the middle point of the base. 17. The sum of the straight lines drawn from any points within a triangle to each of its angles is less than the sum and greater than half the sum of the sides of the triangle. 18. Show that the last exercise is true if a square be substituted for the triangle. 19. Two quadrilateral figures ABCD and AEFD stand on the same base AD and AEPD lies wholly within ABCD. Show that the perimeter of AEPD is less than that of ABCD. [Produce EF to meet AB and CD in G and H.] 20. The shorter diagonal of a rhombus passes through the greater angle. 21. The perpendicular is the shortest straight line which can be drawn from a point to a given straight line, and of all others one nearer to the perpendicular is less than one more remote, and lines equally remote from the perpendicular are equal to one another. Parallel Lines. Props. 27 to 31. 22. A line making equal angles with the sides of an isosceles triangle will be parallel to the base. 23. If A0 bisects the base of an isosceles triangle ABC and a straight line be drawn at right angles to it cutting the equal sides AB, AC in P and Q then PQ shall be parallel to the base, and shall also be bisected by A0. 24. If a line bisecting the exterior angle of a triangle be parallel to the base, show that the triangle is isosceles. 25. On a given straight line to describe a triangle equiangular to a given triangle. 26. Through the angular points of an equilateral triangle straight lines are drawn parallel to the opposite sides. Show that the figure thus formed is an equilateral triangle. 27. From a given point draw a straight line to meet a given straight line and making with it an angle equal to a given angle. 28. From any point o draw lines parallel to the sides of any rectilineal figure, and show that the exterior angles of the figure are together equal to the angles at the point 0, i.e., to four right angles. 29. Prove by aid of the last exercise that each of the angles of an equilateral triangle is equal to two-thirds of a right angle. 30. Every parallelogram which has one of its angles a right angle has all its angles right angles. Value of the Interior and Exterior angles of rectilineal figures. Prop. 32 and corollaries. 31. If one angle of a triangle be equal to the sum of the other two then it is a right angle; but if it be less than the sum of the other two it is an acute angle ; and if greater an obtuse angle. 32. Trisect a right angle. 33. What fraction of a right angle is each of the angies of an isosceles triangle, when the angle at the vertex is four times each of the angles at the base ? Shew how to construct such a triangle. 34. Express as a fraction of a right angle the interior angle of each of the following regular figures, viz. : a hexagon, an octagon, and a decagon. 35. Show that three regular hexagons, four squares, or six equilateral triangles will fill up all the space about a point. 36. If the sides of any hexagon be produced to meet the angles formed by their intersection will be together equal to four right angles. 37. The middle point of the hypotenuse of a right-angled triangle is equally distant from the three angles. 38. The straight lines EC, BD bisect the base angles of an isosceles triangle ABC, and meet the opposite sides in D and E. Show that EC is equal to BD and ED parallel to BC. 39. Describe a triangle of given perimeter and having its angles equal to those of a given triangle. 40. The hypotenuse BC of a right-angled isosceles triangle is trisected in D and E; from D and E the straight lines DF and EG are drawn at right angles to BC and meeting the sides in F and G. Show that DEGF is a square. The Parallelogram. Props. 33 and 34. 41. The diagonals of a rectangle are equal to one another. 42. If the opposite sides of a quadrilateral are equal to one another it is a parallelogram. 43. The diagonals a parallelogram bisect one another. 44. If the opposite angles of a quadrilateral are equal to one another it is a parallelogram. 45. Through a given point draw a straight line so that the part of it intercepted by two given parallel straight lines may be of a given length. 46. ABCD is a trapezium having the side AB parallel to CD. Through ( the middle point of BC, PQ is drawn parallel to AD meeting AB and DC (produced if necessary) in P and Q. Shew that the parallelogram AQ is equal to the trapezium ABCD. 47. Any line passing through the middle point of the diagonal of a parallelogram divides the parallelogram into two equal parts. 48. To bisect a parallelogram by a straight line drawn from a given point in one of its sides. 49. If through the angles of a square straight lines be drawn parallel to its diagonals the figure so formed will be a square having twice the area of the given square. 50. Two of the opposite angles of a parallelogram are joined with the middle points of the opposite sides. Shew that the lines thus drawn will trisect the diagonal of the parallelogram. Parallelograms and Triangles. Equivalent Areas. Props. 35 to 45. 51. Bisect a triangle by a straight line drawn from one of its angles. 52. Bisect a triangle (ABC) by a straight line drawn from a given point (P) in one of its sides (AB). [Bisect BC in 0. Join PO and draw AQ parallel to it. Then P& shall bisect the triangle.] 53. The straight line joining the middle points of two sides of a triangle is parallel to the third side and equal to half of it. 54. The straight lines joining the middle points of the four sides of any quadrilateral form a parallelogram. 55. Bisect a square by a straight line drawn from a given point in one of its sides. (See Exercises 47 and 48.] 56. Divide a square into four equal parts by straight lines drawn from a given point in one of its sides. 57. The diagonals of a parallelogram divide it into four equivalent triangles. 58. If lines be drawn from the angles of a parallelogram to any point within the figure the parallelogram will be divided into two pairs of equivalent triangles. 59. Change a quadrilateral figure into an equivalent triangle. 60. If two sides of a triangle are given the area of the triangle is greatest when they contain a right angle. Relations of the squares described on the sides of a Triangle. Props. 47 and 48. Book i. Props. 12 and 13. Book ii. 61. Make a square which shall be (1) double of a given square. (2) Equal to the squares described on three given lines. 62. Find a square equal to the difference of two given squares. 63. If a perpendicular be drawn from the vertex of a triangle to the base, the difference of the squares on the segments of the base is equal to the difference of the squares on the sides of the triangle. 64. If a point be taken within or without a rectangle the sum of the squares of the lines drawn to one pair of opposite angles is equal to the sum of the squares on the lines drawn to the other two angles. 65. If two right-angled triangles have the hypotenuse, and one side equal in each triangle, the triangles shall be equal -in every respect. 66. An angle of an isosceles triangle is 120° and the sides containing it are each 4 units in length. What is the length of the remaining side ? 67. In any right-angled triangle if a perpendicular be drawn from the right angle upon the hypotenuse the square on this line is equal to the rectangle under the segments of the base. 68. In the last exercise if ABC be the triangle and BD the perpendicular, show that the rectangle CA, AD is equal to the 69. If the base of a triangle be bisected the squares on the other two sides of the triangle are together double of the square on half the base and of the square on the line, joining the middle point of the base with the opposite angle. (Draw AD bisecting side BC of the triangle ABC and AE per. pendicular to it. Then one of the triangles ADC, ADB is obtuse and the other acute. Suppose ADC obtuse. Then by ii. 12 the square on AC=squares on AD, DC increased by the twice the rectangle CD), DE: and by ii. 13 the square on AB=squares on AD.DB. diminished by twice the rectangle BD, DE (=CD.DE). Hence the sqs. on CA, AB=twice the sqs. on CD and DA.] 70. From the last exercise show that the squares on the sides of a parallelogram are together equal to the squares on its diagonals. square on AB. |