5 DEFINITIONS. A point has neither length, breadth, nor thickness. A plane, or superficies, has length and breadth only. A right angle is the inclination of one straight line to another, when, if either be produced, the second angle will be equal to the former. An obtuse angle is greater; an acute angle less, than a right angle. Parallel lines never meet. A parallelogram is a four sided figure, having its two opposite sides parallel. A rectangle is a right angled parallelogram. A square is an equal sided rectangle. A rhombus is equilateral, but not equiangular. A rhomboid is neither equiangular nor equilateral. This A circle is a plain figure bounded by its circumference, every part of which is equidistant from the centre. distance from the centre is called the radius. An arc is a portion of this circumference. The chord of an arc is the straight line, joining the extremities of the arc. A segment is the space included between the chord and the arc. The sector of a circle, is the space included between two radii, and the arc; therefore the sector of a right angle is a quadrant. GEOMETRICAL PROBLEMS. 1. To describe an equilateral triangle upon a given line. Let AB be the given line. From A and B, with the radius AB, describe two circles intersecting in C; join AC and BC. ABC is the equilateral tri angle required. A B 2. From a point, within a given line, to erect a perpendicular. Let AB be the given line, and C be the given point. From C, as a centre, take any distance, CA, and make CB CA. Then from A and B, as centres, at the distance AB, describe arcs intersecting at D; join the point of intersection at D with the point C. CD will be perpendicular to AB. C B 3. At a given point, in a given line, to construct a right angle, or an angle of any number of degrees, by means of a scale of chords. Let AD be the given line, and A the given point. Take off with your compasses (on any scale of chords) AB, the chord of 60 degrees (radius); and from the given point A, with that distance, describe a circle intersecting the given line at B; then, from the point of intersection, with the distance of the chord of 90 degrees E B D BE, or of the chord of any other angle that may be required, BC, on the same scale describe another circle, intersecting the former. Join the points of intersection with the given point A, and the lines will be perpendicular to, or making the required angle, with the given line. A protractor may be used, but not so accurately, for the same purpose. EXAMPLE 1. At the point A, on the line AB, construct angles of 20, 40, 70, 120, 145, 175, and 179 degrees. 3 a. From a point at the end of a given straight line to erect a perpendicular. Let AB be the given straight E line, and A the point at the end. Take any point D and from D as centre, at the dis tance DA, describe the circle EAC, and join CD, and produce to E. Join EA. EA will be the perpendicular required. 4. To bisect a given straight line. Let AB be the given straight line; it is required to bisect it. Upon AB describe the equilateral triangles, ACB and ADB; join the vertices C and D by the line CD, intersecting AB at E. E shall be the point of bisection. A E B 5. To let fall a perpendicular upon a given straight line, from a given point above it. Let AB be the given straight line, and C the given point. From C take any distance CG, and describe the circle EGF, intersecting AB in E and F; bisect EF in H. and join CH. CH is the perpendicular required. 6. To bisect a given angle. Let ABC be the given angle ; take any point D in AB, and from B, as a centre, at the distance BD, describe the circle DGE; join DE, and bisect it in F; join BF. BF shall bisect the angle ABC. A E H G 7. To bisect a given angle, when the inclination of the two sides can only be obtained, and not the vertex of the angle, included between them. Let A and B be the two sides such, that they cannot be produced. Take any points, A and B, and draw g the equal perpendiculars AD and BC; through D and C draw DF and CF parallel to A and B respec D F tively; the angle DFC will be equal and similarly situated to the angle at the vertex. Bisect the angle DFC by the line E F, this line produced will bisect the given angle. 8. To draw a line parallel to a given line. |