For this We can now construct a ▲ on a base 2.3 inches,_having angles of 30° and 65° at the extremities of the base. use the method of Problem VII. Problem IX.-To construct a triangle given two sides and the angle opposite to one of them. Construct a triangle having two sides of lengths 1.4 and 2.4 inches, and an angle of 25° opposite to the shorter side. 25 B C D. K Fig 18. Construct HBK 25° (Fig. 18). From BH mark off BA = 2.4 inches. With centre A and radius 1.4 inches describe an arc cutting BK in C and D. Either of the triangles ABC or ABD may be given as the Δ required. NOTE 1.-The three sides and three angles are sometimes called the six elements of a triangle. We cannot construct a triangle unless we are given three elements, including at least one side. NOTE 2.-In constructing a triangle from any data it is a good practice first to sketch the triangle roughly and mark the given magnitudes. We can then usually see how to construct it accurately with instruments. Examples VI. 1. With a base AB 2" long, draw the triangle ABC whose angles at A and B respectively contain 60° and 30°. Measure the sides AC and BC and the angle ACB. 2. With base AB (3") draw the triangle ABC whose angles at A and B contain respectively 45° and 75°. Measure the sides AC and BC and the angle at C. Construct triangles ABC from the following data, and in each case measure the sides BA, CA and the angle A. Show in every case that the sum of the three angles is 180°: 7. AC 30 mm., = 4. BC= 2", B = 50°, C = 30°. = AB = 30 mm., A = 90°. 8. BC= 40 mm., A = 90°, C = 30°. 9. BC3 cm., A = 120°, C = 15°. 11. AC 10. BC 3", AC = 2′′, A = 70°. 40 mm., AB = 60 mm., C = 120°. 2", AC = 2.5", A 35°. 14. AC 3 cm., AB = 2 cm., C = 40°. In the next four examples M is the middle point of the side BC. 10. On the construction of quadrilaterals.-A figure bounded by four straight lines is called a quadrilateral—for example ABCD (Fig. 19). The lines AC and BD which join opposite corners are diagonals. It will be found that to construct a quadrilateral we must know five elements (using the word "elements to include sides, angles, diagonals, etc.). The problem can be set in many different ways, but at present the student may be content with the simpler cases. Problem X.-To construct a quadrilateral given three sides and the two included angles. Construct a quadrilateral ABCD, such that AB = 1.4 in., BC = 2 in., CD = ·8 in., B = 50°, C = 60°. [If we first make a rough sketch of the quadrilateral we can see how to build it up from the given data. See Fig. 19.] Draw BC2"; make CBA = 50° and arm BA1·4′′; make BCD = 60°, and arm CD=8", Join AD. Problem XI.-To construct a quadrilateral given the four sides and one diagonal. Construct a quadrilateral ABCD such that AB 2 in., BC = 2.5 in., CD = 3 in., DA = 3.5 in., AC = 2·5 in. [If we first make a rough sketch as in Fig. 20 we see that we can construct As ABC and ADC, as we know the three sides of each. Compare Problem I.] = Fig. 20. Draw AC 2.5", with centres A and C and radii 2" and 2.5" respectively describe arcs intersecting at B; with centres A and C and radii 3.5" and 3" respectively describe arcs intersecting at D. ABCD is the required quadrilateral. Problem XII.—To construct a quadrilateral given the four sides and one angle. Construct a quadrilateral ABCD given AB = 1 in., BC= 1.5 in,, CD = 1.2 in., DA = 1·2 in., B = 75°. [See Fig. 21. We can construct A ABC by the method of Problem VI., as we know two sides and the included angle; we can then construct AADC by the method of Problem I.] Draw ABC = 75°, and mark off on its arins lengths BA 1" and BC= 1.5". With centres A and C and radii 1.2′′ describe arcs intersecting at D. A 12" D 75° B 15" C Fig. 21. 12" The student will be able to solve many other simple cases by first drawing a rough sketch, and finding from it how the figure can be built up from the given data. Examples VII. Construct quadrilaterals ABCD from the following data : 1. AB BC 1′′, CD = '3′′, B = 90°, C = = 1.5", BC 1", CD: 1", DA = 90; measure AD. 5 cm., BC = 3 cm., CD 6 cm., DA = 1 cm., ABC=90°; measure ▲ A. = 35 mm., CD 25 mm., DA 2.1", AB 1.4", BC 1", CD = 1", A = 2", BC = 1.5′′, CD =2.2′′, AC = 2.2′′, BD = 50°; measure AC. = = 2·2", measure AD. 35°, DCB 8. BC = 40 mm., ABC = 80°, DBC 9. BD measure AC. = 32 mm., DA = 32 The 11. On parallels.-Learn Definition 30 on page 6. meaning of the word parallel should be easy to understand. The lines drawn on an ordinary sheet of ruled paper form a set of parallel lines; no two of them meet, or would meet even if produced beyond the paper. Parallel lines point in the same direction. Thus if two lines are parallel and one of them points due north, the other will also point due north. Look at the parallel lines in various figures in the book. On page 61, AB is parallel to CD, on page 64, AB, CD, and PQ are all parallel; on page 66, BA is parallel to CE, and on page 73, we have AD parallel to BC, and AB parallel to DC. In Definition 30 page 6, notice the words "in the same plane." The word plane simply means a flat surface;-either a real flat surface, such as the top of a table, or an imaginary flat surface such as we can conceive stretched across between the posts of the goal at football. Now it is quite possible to draw two lines which do not meet when produced, and yet which are not parallel. For instance draw a line on the table pointing north and south, and a line on the floor pointing east and west. These two lines would not meet when produced, and yet are not parallel, for they point in quite different directions. But these lines are not in the same plane, for you could not imagine a flat surface which contained both lines. Thus we cannot assert that two lines are parallel unless (i) they would never meet, and (ii) they lie in the same plane. We shall frequently use the symbol || for parallel. Problem XIII.—To draw a line parallel to a given line at a given distance. Draw a line || to AB at a distance of '8". Take any two points in the line H and K. With centres H and K and radius 8" describe two arcs on the same side of the line. Draw CD touching each of these arcs (as shown in the figure). will be the required parallel. CD A H K B Fig. 22. Every point in the line CD will be at a distance 8" from the line AB. Note that the distance of a point from a line is the length of the perpendicular drawn from the point to the line. E Problem XIV.—To draw a line through a given point parallel to a given line. Take any convenient point C in AB. Alternative Method.-Place one set square so that one side lies along AB, as in the position MCD (Fig. 24). Place the other set square against H H A B Fig. 24. it, in the position EFG. Keeping EFG fixed, slide the first set square along it till the side CB reaches the point P, as in the position HKL. Rule the line HPK, which will be the required parallel. We may also draw the parallel by using parallel rulers. Examples VIII. 1. Take a line AB of length 2.5". Bisect it at C. Draw lines to AB at the points A, C, and B respectively. Note that these lines are parallel. Join any point H on the first perpendicular to any point K on the third, and show by measurement that HK is bisected by the second perpendicular. 2. Take a line AB of length 2.4". Bisect it at C. Construct an angle 40°. Through C draw a line CE || to AD. From B draw a line 1 to AD meeting AD in F and CE in G. Measure the angles BCE and BGE, and the lengths BG, GF. BAD 3. Draw three parallel lines. Draw a line cutting all three lines and making an angle of 50° with one of them. Show that every angle in the figure measures either 50° or 130°. 4. Draw a line, a second parallel to it, and 1 inch above; a third parallel to the second, and 2 inches above. Draw a line making an angle of 42° with the lowest line, and cutting the other lines. Measure all the angles in the figure. 5. Draw a line, and two other lines parallel to it. angles to one of the lines and cutting the other lines, angles in the figure. == Draw a line at right and measure all the = 35°, and an angle Note that AC is to Measure AO, 0. 6. Draw AB of length 1.5". Make an angle BAC : ABD 35° (AC and BD being on opposite sides of AD). BD. Make AC BD = 1·7". Join CD, cutting AB in BO, CO, DO. Note that AB and CD bisect one another. 7. Construct an equilateral triangle of side 4 cm. draw a line || to the base. Measure the distance of any point on this parallel from the base. 8. Draw a triangle having sides 30, 40, 50 mm. Through the vertex Draw a line | to the longest side through the opposite vertex of the triangle. Measure the distance between these parallels. (i) If a line is perpendicular to one of a set of parallel lines, it is perpendicular to them all. (Compare question 5.) (ii) If a line cut two or more parallel lines, and is not perpendicular to them, all the acute angles formed are equal, and all the obtuse angles formed are equal, and any acute angle together with any obtuse angle makes up 180°. (Compare question 4.) |