fig. 37. fig. 38. fig. 39. Ex. 80. Copy fig. 37, taking 5 cm. for the radius of the circle. Where are the centres of the arcs? A straight line drawn through the centre of a circle to meet the circumference both ways is called a diameter. The two parts into which a diameter divides a circle are called semicircles. Ex. 81. Copy fig. 38, taking AD = 9 cm. AD is a diameter of a circle and is divided into three equal parts at B and C; semicircles are described on AB, AC, CD, BD as diameters. Ex. 82. Draw a figure showing the points of the compass. Ex. 83. Copy fig. 39, taking 5 cm. for the radius of the large circle. The radius of the small circle is half that of the large circle; the centres of the arcs vertices of the regular hexagon. are the fig. 40. Ex. 84. Copy fig. 40. The points of the star are the vertices of a regular pentagon. TRIANGLES. Ex. 85. Draw a triangle (each side being at least 2-5 in. long). Measure all its sides and angles; find its perimeter and the sum of its angles. Ex. 86. Repeat Ex. 85 three or four times with triangles of different shapes. When told to construct a figure to given measurements, first make a rough sketch of the figure on a small scale and write the given measurements on the sketch. Ex. 87. Make an angle ABC=74°; cut off from its arms BC = 3.2 in., BA = 2·8 in.; join AC. In all cases where triangles or quadrilaterals are to be constructed to given measurements, measure the remaining sides (in inches if the given sides are measured in inches, in centimetres if the given sides are measured in centimetres); also measure the angles, and find their sum. Construct triangles to the following measurements:— 4 ABC = 80°, AB = 2.2 in., Ex. 88. (i) BC= 2.9 in. Ex. 89. to cut at C. Draw a straight line AB 9 cm. long, at A make an angle BAC = 60°, at B make an angle ABC = 40°, produce AC, BC Measure the remaining sides and angle of the What is the sum of the three angles? triangle ABC. Ex. 90. Construct triangles to the following measurements:— (In case the construction is impossible with the given measurements, try to explain why it is impossible.) Ex. 91. (i) (ii) CA = 9.0 cm., Ex. 92. Construct triangles to the following measurements:- Construct quadrilaterals ABCD to the following measurements: (Here it is especially important that, before beginning the construction, a rough sketch should be made showing the given parts. Note that the letters must be taken in order round the quadrilateral; e.g. the quadrilateral in fig. 41 is called ABCD and not ABDC.) A B (v) B=122°, ▲c=130°, ▲ D=130°, BC = CD = 1.6 in. (vi) AD=3.0 in., AB = 2.4 in. D=118°, 4 DAC 27°, 4 BAC = 35°, (vii) AC=5.6 cm., ▲ BAC = 58°, ▲ DAC = 69°, 4 BCA = 58°, ▲ DCA = 69°. (viii) AB = 1.9 in., BD = 1.7 in., CD 2.0in., 4 ABD=118°, < BDC = 23°. (ix) ABCD = 5.8 cm., < BDC=46°. AD = 4.7 cm., LA=72°, (x) AB=6.3 cm., CD=5.4 cm., 4 BAC = 64°, ▲ ACD = 59°, LD=76°. (xi) AB=5.2 cm., AC = 6.8 cm., AD=5'6 cm., ▲ BAC=106°, ▲ BAD = 122°. Ex. 93. Take a point O on your paper and mark a number of points each of which is 2 in. from O. [To do this most easily, open your dividers 2 in., place one point at O, and mark points with the other.] The pattern you obtain is a circle; all the points 2 in. from O are on this circle. Ex. 94. How does a gardener mark out a circular bed? Ex. 95. Draw a figure to represent the area commanded by a gun which can fire a distance of 5 miles in any direction. (Represent 1 mile by 1 cm.) Ex. 96. Two forts are situated 7 miles apart; the guns in each have a range of 5 miles; draw a figure showing the area in which an enemy is exposed to the fire of both forts. (Represent 1 mile by 1 cm.) Ex. 97. A circular grass plot 70 feet in radius is watered by a man standing at a fixed point on the edge with a hose which can throw water a distance of 90 feet; show the area that can be watered. (Represent 10 feet by 1 cm.) What is the distance between the two points on the edge of the grass which the water can only just reach? Ex. 98. Mark two points A, B, 3 in. apart. (i) On what curve do all the points lie which are 2·7 in from A? (ii) On what curve do all the points lie which are 2-2 in. from B? (iii) Is there a point which is 2.7 in. from A and also 2.2 in. from B? (iv) Is there more than one such point? Ex. 99. A and B are two points 7.4 cm. apart; find, as in Ex. 98, a point which is 57 cm. from A and 3.5 cm. from B. Ex. 100. Repeat Ex. 99, without drawing the whole circles. See fig. 42. Ex. 101. (i) Construct a triangle, the lengths of whose sides are 12·1 cm., 8.2 cm., 6·1 cm. See Ex. 100. (ii) In how many points do your construction circles intersect? (iii) How many triangles can you construct with their sides of the given lengths? Are these triangles congruent (i.e. could they be made to fit on one another exactly)? Ex. 102. Construct triangles to the following measurements :— (It is best to draw the longest side first.) |