Problems on Propositions 1-34. 7. Verify Euclid I. 24 in the case of two triangles each of which has two sides of 30 and 50 mm. length respectively, the included angle in the first being 90° and in the second 135°. = 8. Draw a triangle ABC such that AB = 2", BC = 2.5′′, CA 3". Bisect AB and AC at M and W respectively. Draw AD 1 BC and meeting BC in D; also draw ME and NF 1 BC. Now cut the triangle out with a penknife. Fold each of the corners A,B,C to the point D. (The creases of the paper will be MN, ME, and WF). Hence show that the angles A, B, C are together equal to two right angles (as they are equal to the three angles MDN, MDE, NDF respectively). This construction will hold for any triangle, but if the triangle is obtuse BC must be the longest side. 9. Draw the locus of all points which are 7" distant from a given line AB (of indefinite length). [The locus consists of two lines | to AB, one on each side.] 10. Draw two lines AOB, COD intersecting at O, such that AOC = 70°. Draw the locus of all points which are 6" from AOB, and the locus of all points which are '9" from COD. Mark 4 points, each of which is 6" from AOB and 9" from COD. 11. Draw two lines AOB, COD as in Question 10. Draw the locus of all points which are equidistant from these lines. (The locus consists of the lines bisecting each of the angles formed at 0.) = 3 cm., BC = 4 cm., CA = 5 cm. 12. Draw a triangle ABC, such that AB Find two points, one in CB and one in CB produced, which are equidistant from AB and AC. Measure the distance of each point from AC. 13. Draw a triangle ABC as in question 12. Find a point within the triangle equidistant from all three sides. Measure its distance from either side. 14. Draw an acute-angled triangle on a base 2", of altitude 1", having one side 1.3". Measure the other side. = 15. Draw an obtuse-angled triangle, having the measurements given in question 14. Measure the other side. Problems on Propositions 1-48. 16. Show by measurement that if the sides of a right-angled triangle are 3 and 4 cm. respectively, its hypotenuse is 5 cm. [Cf. Prop. 47.] 17. Show by measurement that the triangle whose sides are 5", 12", 1.3" respectively is right-angled. [Cf. Prop. 48.] 18. Show by measurement that the triangle whose sides are 5", 12", 1.4" respectively is obtuse-angled. [Cf. question 17, and Prop. 25.] 19. Show by construction that the area of a rectangle whose adjacent sides are 3 and 5 cm. is 15 square centimetres. 20. Show by construction that the area of a rectangle whose adjacent sides are" and "is square inch. 21. Show by construction (using question 20) that the area of a rectangle whose sides are 23" and 23" is thirty-five sixths of a square inch. 22. Assuming that the area of a rectangle is in all cases given by the product of two adjacent sides (compare questions 19, 20, 21), prove that base x altitude. [Cf. Prop. 35.] (i) area of a parallelogram = (ii) area of a triangle = base × altitude. [Cf. Prop. 41.] (iii) area of a right-angled triangle = product of the sides containing the right angle. [Cf. Prop. 34.] 23. Construct again the first four triangles given on page ix and calculate their areas. 24. Construct again the first four quadrilaterals given on page xx; reduce each to a triangle by the method of page 83, and thence calculate their areas. 25. Draw a line 7 cm. long. If this line is the base of a triangle whose area is 14 sq. cm., draw the locus of the vertex. = 26. Construct on isosceles triangle having a base of = 3.6" and an area = 4.32 sq. inches. Measure the sides. [First calculate its altitude.] = 2-2", an angle = 70°, and 28. Construct a triangle having an area of 15 sq. cm., an angle = 50°, and one side adjacent to this angle = 6 cm. Measure the side opposite to this angle. 29. Construct a triangle in which two sides measure 4 and 4.5 cm. respectively, and the area is 7 sq. cm. Show that from these data two different triangles can be drawn, and measure the third side in each case. 30. Construct a rectangle of sides 20 and 30 mm. Construct another rectangle of equal area having one side 16 mm. [Use Prop. 44.] Measure the other side and verify by calculation. BE 31. If in the figure of Prop. 44 (p. 86), LGBE = 90°, BG = p inches, 9 inches, BA = r inches, prove that BM = px q÷r inches. = 32. Use the result of question 31 to evaluate the following results by geometrical construction. [Note that we need not reproduce the whole figure of Prop. 44. To evaluate pxq÷r;-make HG = r, GB = p, = LHGB = 90°; draw BE q and to HG; draw EK || to GB meeting HB produced at K; measure EK.] (i) 16 × 30÷20 [work in mm.]; (ii) 1·5×3÷2·4 [work in inches]; (iii) 2.1×3.2 [work in cm. and make r = 1]; (iv) 2·3÷1·3 [work in inches and make p = 1]; (v) 1·5x4·1÷3·2; (vi) 2.3×34; (vii) 4.5÷1.8; (viii) 7·3÷4·2. 33. Construct a rectangle whose sides are 1.3′′ and 2.3′′; construct a parallelogram of equal area whose sides are 1.6" and 2.6". Measure the acute angle of the parallelogram. [Use Prop. 35.] 34. Draw a right-angled triangle in which the hypotenuse is 2" and one side is 1". Find the length of the other side by measurement and calculation. 35. Draw a square equal to the sum of two squares whose sides are respectively 1", 2". Measure its diagonal. 36. Verify the last result by calculation. 37. Draw a square equal to the difference of two squares whose sides are 22 and 33 mm. Measure its diagonal. 38. By means of Prop. 47 construct lines of the following lengths and measure them—(i) √/13′′, (ii) √/5′′, (iii) √41 mm., (iv) √29 mm., (v) √/7", (vi) √23 cm., (vii) √33 cm., (viii) √43 cm. 39. Construct a triangle whose perimeter is 4 inches, and whose angles are 40°, 60°, 80°. Measure its longest side. [See Ex. 11, p. 102.] 40. Calculate algebraically from Book I. the perimeter of an equilateral triangle whose area is 2 sq. cm. 41. Calculate algebraically from Book I. the perimeter of a triangle whose sides are in the ratios 3: 4: 5, and whose area is 1 sq. mm. 42. Using question 8, show that the area of a triangle is double that of a rectangle contained by half the height and half the base of the triangle. BOOK II. ON RECTANGLES AND SQUARES. DEFINITIONS. 1. A rectangle is a parallelogram which has one of its angles a right angle. 2. A rectangle is said to be contained by any two of its sides which contain a right angle. 3. In any parallelogram the figure formed by either of the parallelograms about a diagonal together with the two complements is called a gnomon. The shaded portion in the figure is a gnomon. NOTES ON THE DEFINITIONS. 1. A rectangle is often defined as a right-angled parallelogram; but if one angle of a parallelogram is a right angle, all its angles are right angles, and therefore it is sufficient to know that one angle is a right angle. 2. In speaking of rectangles the words 'contained by' are often omitted; e. g. 'the rectangle AB, CD' or 'the rectangle under AB, CD' is taken to mean 'the rectangle contained by AB and CD.' It is evident that both the shape and area of a rectangle are known if the lengths of two sides which contain a right angle are given; and the rectangle AB, CD is the same as the rectangle CD, AB. In Arithmetic and Algebra the area of a rectangle is represented by the product of the length into the breadth of the rectangle; and the propositions of Euclid, which deal with rectangles, can generally be represented either arithmetically or by means of algebraical formulæ. Many writers also use various algebraical signs and abbreviations in propositions belonging to Pure Geometry. These are useful, because they are brief, and they may perhaps be used in Riders; but they should not be used in the ordinary propositions of Euclid. The following are examples of such signs and abbreviations : AB.CD means the rectangle contained by AB and CD. There is, however, an important difference between problems and theorems in pure Geometry and those in Arithmetic and Algebra. In the latter, lengths, areas, etc. are always expressed in terms of some unit, as e. g. an inch, a square foot, etc.; but in Pure Geometry no such units are used. = B, may be The length of any line, as e. g. A. measured in two directions. We may start from A and measure AB; or we may start from B and measure BA. In works on Mathematics it is usual to express this difference by the signs + and Lines which are measured horizontally from left to right are considered positive; and lines measured in the reverse order are considered negative. This distinction is also observed in naming a line, so that AB BA; and the rectangle AQ, QB rectangle AQ, BQ. This convention is of less importance in Euclid; but it is as well for the learner to be taught to observe it from the first, and so the propositions in this edition of Euclid are printed according to the rule, except in Prop. 6 of the second book. There the rule has been disregarded in order to make clear to the beginner the similarity of Prop. 6 to Prop. 5. Of course, this convention about signs does not affect squares, because all real square quantities are positive; and AB2 = BA2. PROP. 1.-Theorem.-If there be two straight lines, one of which is divided into any number of parts, the rectangle contained by the two straight lines is equal to the sum of the rectangles contained by the undivided line and the several parts of the divided line. Let X and AB be two straight lines, and let AB be divided into any number of parts AC, CD, DB; then the rectangle contained by X, AB shall be equal to the sum of the rectangles contained by X, AC, by X, CD, and by X, DB. |