43. Divide a given straight line into two parts so that the difference of the squares on the parts may be equal to a given rectilineal figure. When is this impossible? 44. Divide the straight line AB at the points C and D, so that, if AC be equal to DB, the square on AC may be equal to the rectangle AB, CD. 45. Divide a given straight line into two parts so that the sum of the squares on the whole and one part may be equal to (i) twice, (ii) three times, (iii) four times, or (iv) five times the square on the other part. 46. Find the length of the side of a square equal in area to a hombus whose side is three inches long and one of whose angles is 150°. 47. Divide a straight line into two parts so that the rectangle contained by the two parts may be equal to a given square. Is this always possible? 48. Describe a rectangle equal to a given rectilineal figure and having its perimeter a minimum. 49. Describe a rectangle equal to a given square and having the difference of two of its adjacent sides equal to a given straight line. 50. Describe a square equal to a given regular pentagon. 51. Describe a square whose area shall be equal to the sum of the areas of the squares described on the three sides of a given right-angled triangle. 52. Describe a right-angled triangle having one of its sides double of the other and equal in area to a given rectilineal figure. 53. Describe a parallelogram equal in area to a given square, having its perimeter equal to a given straight line, and having an angle equal to a given angle. When is this impossible? 54. Find the locus of a point, such that the sum of the squares on its distances from three given fixed points is constant. 55. ABC is any triangle; find a point P within it such that the sum of the squares on AP, BP and CP is a minimum. 56. Find the position of a point such that the sum of the squares on its distances from four given fixed points is a minimum. 57. Given the base of a triangle and the sum of the squares on its sides, find the locus of its vertex. 43. Divide a given straight line into two parts so that the difference of the squares on the parts may be equal to a given rectilineal figure. When is this impossible? 44. Divide the straight line AB at the points C and D, so that, if AC be equal to DB, the square on AC may be equal to the rectangle AB, CD. 45. Divide a given straight line into two parts so that the sum of the squares on the whole and one part may be equal to (i) twice, (ii) three times, (iii) four times, or (iv) five times the square on the other part. 46. Find the length of the side of a square equal in area to a rhombus whose side is three inches long and one of whose angles is 150°. 47. Divide a straight line into two parts so that the rectangle contained by the two parts may be equal to a given square. Is this always possible? 48. Describe a rectangle equal to a given rectilineal figure and having its perimeter a minimum. 49. Describe a rectangle equal to a given square and having the difference of two of its adjacent sides equal to a given straight line. 50. Describe a square equal to a given regular pentagon. 51. Describe a square whose area shall be equal to the sum of the areas of the squares described on the three sides of a given right-angled triangle. 52. Describe a right-angled triangle having one of its sides double of the other and equal in area to a given rectilineal figure. 53. Describe a parallelogram equal in area to a given square, having its perimeter equal to a given straight line, and having an angle equal to a given angle. When is this impossible? 54. Find the locus of a point, such that the sum of the squares on its distances from three given fixed points is constant. 55. ABC is any triangle; find a point P within it such that the sum of the squares on AP, BP and CP is a minimum. 56. Find the position of a point such that the sum of the squares on its distances from four given fixed points is a minimum. 57. Given the base of a triangle and the sum of the squares on its sides, find the locus of its vertex. PRACTICAL EXERCISES ON BOOK II. 1. Show that Prop. 7 is equivalent to the algebraic formula (a−b)2 = a2+b2 — 2ab. 2. Show that Prop. 8 is equivalent to the algebraic formula 3. In the figure of Prop. 14 it follows from Euclid's arguments that BE. EF. Use this figure to construct a line of length 14 cm. 2 cm.] EH2 = measure the line. [Make BE 7 cm., EF = [Note that we need not construct the rectangle BD.] m.and 4. Construct a line of length √3.3′′ and measure it. [Make BE = 3′′, EF = 1·1".] 5. Using the figure of Prop. 14 construct and measure lines of the following lengths:—(i) √7′′, (ii) √10 cm., (iii) 19 cm. [195×3·8], (iv) √23 cm., (v) 4.5 in., (vi) √34 in. 6. Describe a rectangle whose perimeter is 6 square inches. Measure its sides. inches and whose area is 1.5 [Using the figure of Prop. 14:-Make BF 3"; describe the semicircle; make FK 1.5" and 1 to BF; draw KH || to BF, meeting the circumference in H; draw HE L to BF; construct the rectangle BEDC.] = 7. Describe a rectangle whose area is 9 sq. cm. and whose perimeter is 14 cm. Measure its sides. 8. Find geometrically two numbers whose sum is 7 and whose product is 9. [Cf. question 7.] 9. Find geometrically two numbers whose sum is 20 and whose product is 70. [Work in tenth-inches.] 10. Show how to produce a given line to such a length that the rectangle contained by the whole line and the part produced shall be equal to the square on a given line. [Use II. 6 and I. 47.] Hence find geometrically two numbers whose difference is 4 and whose product is 20. 11. Use Add. Prop. 5, p. 148, to calculate the length of the median of a triangle if the base is 3 inches, and the sum of the squares of the sides is 12 square inches. 12. Draw the locus of the vertex of a triangle on a given base of length 3 inches, if the sum of the squares on the sides is 12 square inches. [Cf. question 11]. 13. Construct a triangle in which the base is 40 mm., the area is 400 sq. mm., and the sum of the squares on the sides is 2000 sq. mm. Measure the shorter side. 14. Draw a rectangle whose sides are 2′′ and (3/5)". Prove graphically that it is equal to the square on a line of (√5−1)". = [In practical work, the construction of Euclid II. 11, p. 140, is modified as follows:-Draw BE AB and 1 to AB; join EA; from EA mark off EL = EB; from AB mark off AH AL; then AH2 = AB. HB. this is equivalent to Euclid's construction. Prove by showing that practical work, the construction of Add. Prop. 10, p. 152, is modified as follows:-Draw BE = AB and to AB; join EA; on AE produced mark off EL = EB; on BA produced mark off AK == AL; then AK2 = AB. BK. Prove by showing that this is equivalent to the construction on p. 152. Note also the following construction :-Bisect AB at P; draw BQ = AB and 1 to B; with centre P and radius PQ describe an arc cutting AB produced in R, then AB2 = AR.RB. Prove by Euclid II. 6 and I. 47.] 1 Divide a line of length 3" in medial section; also divide the larger portion in medial section. Measure the parts. [Cf. Add. Prop. 11, p. 152.] 16. Take a line of length 2" and divide it externally in medial section. Measure the produced part of the line. [Cf. Add. Prop. 10, p. 152.] 17. Using the figure and the result of Prop. 12, calculate the length of CD if AB = 6", BC= 3", CA = 4". 18. Using the figure and the result of Prop. 13, calculate the length of CD if AB = 4", BC = 5′′, CA = 6′′. 19. Draw a line AB of length 6 cm. Draw the locus of a point such that the sum of the squares of its distances from A and B is 25 sq. cm. 20. Draw a line AB of length 6 cm. Draw the locus of a point such that the difference of the squares of its distances from A and B is 12 sq. cm. Add. Prop. 4, p. 152.] [Cf. 21. Draw a line AB of length 6 cm. Find a point such that the square of its distance from A exceeds the square of its distance from B by 18 sq. cm., and such that its perpendicular distance from the line AB is 4 cm. Measure its distance from B. 22. Using II. 13 and I. 47 calculate the altitude of a triangle whose base is 21", and whose sides are 20′′ and 13′′. 23. Calculate the lengths of the medians of a triangle of sides 8′′, 12′′, 16′′. [Cf. Add. Prop. 5, p. 152.] 24. In a certain triangle the sum of the squares on the sides is 6800 square inches, and the median and altitude measure 50" and 40′′ respectively. Calculate the lengths of the sides. EUC. I., II. M |