CONSTRUCTIONS. Hitherto we have constructed our figures with the help of graduated instruments. We shall now make certain constructions with the aid of nothing but a straight edge (not graduated) and a pair of compasses. We shall use the straight edge (i) for drawing the straight line passing through any two given points, (ii) for producing any straight line already drawn. We shall use the compasses (i) for describing circles with any given point as centre and radius equal to any given straight line, (ii) for the transference of distances; i.e. for cutting off from one straight line a part equal to another straight line. [(ii) is really included in (i).] By means of theorems which we have already proved, we shall show that our constructions are accurate. In the exercises, when you are asked to construct a figure, you should always explain your construction in words. You need not give a proof unless you are directed to do so. In the earlier constructions the figures are shown with given lines-thick, construction lines-fine, required lines of medium thickness, lines needed only for the proof-broken. In making constructions, only the necessary parts of construction circles should be drawn even though "the circle" is spoken of. Revise Ex. 98-102. To construct a triangle having its sides equal to three given straight lines. Let X, Y, Z be the three given straight lines. Construction Draw a straight line PQ. From PQ cut off a part PR X. With centre P and radius = Y describe a circle. Z describe a circle. Let the circles intersect at S. Join PS, RS. Then PRS is the required triangle. NOTE. It is best to draw the longest line first. It should be observed that the construction is impossible if one of the given straight lines is greater than the sum of the other two. Ex. 501. Ex. 502. of fig. 8. (Why?) * Draw a large triangle and construct a congruent triangle. Construct a triangle having its sides equal to the lines b, d, h Ex. 503. Draw a straight line (about 3 in. long); on it describe an equilateral triangle. Measure its angles. Ex. 504. Construct an isosceles triangle of base 5 cm. and sides 10 cm. Measure the vertical angle. Ex. 505. Draw an angle ABC; complete the parallelogram of which AB, BC are adjacent sides. [On AC construct ▲ ACD=▲ CAB.] Give proof. Ex. 506. Make an angle of 60° (without protractor or set square). Ex. 507. Make an angle of 120° (without protractor or set square). Revise Ex. 274-276. * Constructions should always be made on a large scale; an error of 5 mm. is less important in a large figure than in a small one. In this case let the shortest side be at least 3 in. long. Through a point o in a straight line ox to draw a straight line OY so that given angle BAC. XOY may be equal to a VA A fig. 131. X Construction With centre A and any radius describe a circle cutting AB, AC at D, E respectively. With centre O and the same radius describe a circle PY cutting OX at P. With centre P and radius = DE describe a circle cutting the circle PY at Y. [The protractor must not be used in Ex. 508–518.] Ex. 508. Draw an acute angle and construct an equal angle*. Ex. 509. Draw an obtuse angle and make a copy of it. Ex. 510. Draw an acute angle ABC; at C make an angle BCD = LABC. Let BA, CD intersect at O. Measure OB, OC. Ex 511. Draw a triangle ABC; at a point O make a copy of its angles in the manner of fig. 50. Ex. 512. Repeat Ex. 511 for a quadrilateral. Ex. 513. Draw two straight lines and an angle. Construct a triangle having two sides and the included angle equal respectively to these lines and angle. Ex. 514. Ex. 515. Construct a triangle ABC having given BC, LB and LC. Construct a triangle ABC having given BC, LA and LB. Ex. 516. Draw a straight line EF and mark a point G (about 2 in. from the line); through G draw a line parallel to EF. [Draw any line through G cutting EF at H; make HGC-4GHF; see fig. 86.] Ex. 517. Repeat Ex. 516, using corresponding instead of alternate angles. Ex. 518. Draw a large polygon and make a copy of it, using the first method described on p. 50. Revise "Symmetry" pp. 51-55. Ex. 519. Cut out an angle of paper; bisect it by folding as in Ex. 31. * It is convenient to draw the angle on tracing paper so as to compare it with the angle made equal to it. Construction From AB, AC cut off equal lengths AD, AE. With centres D and E and any convenient radius describe equal circles intersecting at F. Join AF. 'Any convenient radius." If it is found that the equal circles do not intersect, the radius chosen is not convenient, for the construction breaks down; it is necessary to take a larger radius so that the circles may intersect. |