55. Draw a vertical line AB, 2 inches long. Let this be a diameter of a square. Construct the square, and then divide it into nine smaller squares, equal to each other.

56. About a given circle describe a triangle having angles of 20°, 60°, and 75° respectively.

57. On a given line AB, 2 inches long, construct a regular hexagon, by the use of the set-square of 60°.

58. Draw three lines 14 inch, 2 inches, and 2 inches respectively, and find their fourth proportional.

59. Draw a rectangle of 4 inches and 24 inches sides. Within it inscribe an ellipse, which shall touch the centre of each side tangentially.

60. Find the length of AB, the mean proportional between two lines, 1 inch and 3 inches long.

(F.)

61. Draw a square, and by means of parallels to its sides-at a distance of inch--construct another square.

62. Draw a scale to represent 15 feet, on a scale of §th of an inch to an inch.

63. Let a line AB (6 inches) represent the plan of one side of a street. Show the plan of the opposite side, which is to be parallel to AB. The street is to be 15 feet wide. [Scale—1 inch to 10 feet.]

64. Draw a line 4.5 inches in length, and at one extremity erect a perpendicular 1.5 inch long. From the top of the perpendicular draw a line making an angle of 30° with the given line.

65. Draw the plan of a circular race-course 1 mile in diameter, so that three gates, A, B, and C, shall fall within the path. [Scale1 inch to a mile.]

66. Draw a horizontal line AB, 2 inches in length. Let this be the base of a triangle having one side the length of AB, and the other side of AB. Construct the triangle.

67. Draw a square of 2.5 inches side, and inscribe another within it, having each of its corners in the sides of the first, and at 1 inch from its angular points. Describe the circles circumscribing these two squares.

68. From a given point A, 11⁄2 inch outside the circumference of a given circle whose diameter is 2 inches, draw a tangent to the circle.

69. An isosceles triangle has one of the angles at its base equal to 36° 14′ 23′′, what will be the vertical angle?

70. A line AB is 35 inches long. Divide it in the point C, so that AB: BC:: 7:4.

71. Describe a circle which shall pass through three consecutive angles of any regular nonagon.

72. The diagonal of a rectangular drawing-board is 24 feet. One of the sides of the board makes an angle of 30° with one end of the diagonal. Draw the plan of the board. [Scale-1 inch to the foot.]

(G.)

73. Divide a line 6 inches in length in extreme and mean proportion, and prove by construction that the greater segment is a mean proportional between the whole line and the less segment.

74. Inscribe in any given circle, that polygon whose angles at the centre are each 60°.

75. Draw any irregular polygonal figure, say an irregular pentagon. Let this represent the plan of a field, and draw another plan similar and equal to it.

76. Construct a triangle, two of its sides being 3 inches and 2 inches respectively, and the included angle 50°.

77. About any given circle, describe a regular pentagon, whose sides are parallel to an inscribed pentagon.

78. Construct a rectangle, making one of the sides of the length of the adjacent side.

79. Place two equal lines, 1 inch long, at any angle of 135°. Consider them as two sides of a polygon, and complete the figure.

80. In any scalene triangle whose base is 2 inches, inscribe a rectangle having its base 1.5 inch.

81. Describe a circle of 2 inches diameter. Inscribe within it an irregular polygon, having angles at the centre, equal respectively to 45°, 60°, 30°, 105°, and 120°.

82. Construct a triangle ABC, having its angles 50°, 60°, and 70°, and circumscribing a circle of 1 inch radius.

83. Describe a tre-foil and quatre-foil having adjacent diameters of of an inch.

84. State how many degrees are contained in the angles of each of the following regular polygons—a hexagon, an octagon, and a do-decagon.

(H.)

85. Draw a pentagon, having its side 1 inch in length. Divide it into five isosceles triangles, by drawing lines from its centre to the angular points, and inscribe a circle in each,

86. Construct a quatre-foil and cinque-foil, having tangential arcs, the radius of which is inch.

87. On a given line AB, 1 inch in length, construct a regular octagon, with a set-square of 45o.

88. Bisect a triangle, having its sides 4.5, 5, and 5.5 inches, by a line drawn perpendicular to the longest side.

89. Construct an ellipse in two different ways, the transverse diameter AB, and conjugate diameter CD, being given.

90. Draw a sector of 3 inches radius, and containing an angle of 120°. Divide it into six smaller sectors, all equal to one another. 91. Make any irregular figure of six sides, and construct an equilateral triangle equal in area.

92. The base of a scalene triangle is inch. Its angles are respectively 40°, 60°, and 80°. Describe a similar triangle, whose base is 1 inch.

93. Draw an oblique line AB, 4 inches in length, and let it represent the major axis of an ellipse. Then draw the minor axis a length of 1 inch. Describe the curve of the ellipse, by means of intersecting arcs, and draw a tangent through any point C in the

curve.

94. Describe two circles equal to the sum and difference respectively of two other circles of 1.5 inch and 3 inches diameter.

95. From a circle, whose radius is 1ğ inch, cut off a segment which shall contain an angle of 50°.

96. Find the mean proportional to two lines AB and CD, being 3 inches and 2 inches respectively in length.

(I.)

97. Construct a pentagon having a diagonal 4 inches in length, and a square equal to it in area.

98. The perimeter of a triangle is 6 inches. Construct it so that its sides are in the proportion of 2, 3, and 4.

99. About a circle of 3 inches in diameter, construct a triangle having an angle of 30° and another angle of 45°.

100. The centres of two circles are 2.5 inches apart, having their radii respectively inch and inch. Draw the four lines touching both circles.

101. Draw an equilateral triangle having a base of 2 inches, and construct a rectangle equal to it in area.

102. Within a square of 3 inches side, inscribe the largest possible equilateral triangle.

103. A line 17.5 yards long is represented on a certain drawing by 3:5". Construct the scale to show yards.

104. Construct a rhombus having a base of 4 inches, and two angles of 45°, and make a triangle of equal area having one angle of 70°.

105. From any point C in the circumference of a circle of 14 inch radius, draw a chord which will cut off a segment containing an angle of 30°.

106. Determine of an inch by diagonal division.

107. Draw a rhomboid on a base of 1·5 inch, and construct an isosceles triangle of equal area.

108. On a given line AB, 2 inches long, construct an equilateral triangle. On the same line construct a scalene triangle, of the same area as the equilateral triangle, and having one of its angles equal to 20°.

(K.)

109. Construct an equilateral triangle equal in area to a square, the base of which is 1 inch.

110. Construct a square of 1 inch side. Then on one of the sides construct an isosceles triangle equal to the square in area.

111. Construct a triangle equal in area to two similar triangles, the bases of which are respectively 1 inch and 11⁄2 inch.

On a

112. Draw any irregular seven-sided rectilineal figure. given line AB, 21⁄2 inches long, construct a rectangle equal in area to the irregular figure.

113. Construct a triangle equal in area to the difference between two similar triangles, the bases of which are respectively 2 inches and 1 inch.

114. Draw a regular pentagon of 1 inch sides. Then construct the following figures, each having the area of the pentagon ;-viz. a square, a right-angled triangle, and a rhomboid, the latter containing acute angles of +5°.

115. Describe a circle having a radius of 1 inch, and construct a rhomboid equal to it in area, and having an angle of 45° at the base.

116. Within an equilateral triangle of 2 inches sides, insert a trefoil of tangential arcs of circles.

117. Draw a circle having a radius of inch, and a line AB at any distance from it. Draw a circle which shall touch both the given line and circle.

118. Draw a trapezium having adjacent pairs of sides equal, and respectively 2 and 3 inches in length. Within the trapezium, inscribe a circle and a square.

Show

119. Two lines, AB and CD, converge towards each other. how the angle at which they meet may be bisected when it is inaccessible.

120. Draw a sector containing an angle of 120°, and having radii of 1 inch. Inscribe a circle within this sector, which shall touch the arc and the radii, tangentially.

END OF PLANE GEOMETRY EXERCISES.