PROBLEM XVII. E To describe a rectangle, whose length and breadth shall be, respectively, equal to two given lines, AB and C. At the point B, in the given line AB, erect BD=C. From the point D, with a radius AB, describe an arc, and from the centre A, with a radius C, describe another arc, A! cutting the former in E; join ED and AE: the figure so formed will be the rectangle required. = B ୯ NOTE. Any parallelogram may be described in nearly the same manner, by drawing BD so as to make the proper oblique angle. This may be effected, by taking the chord of 60° from a scale of chords, and from the centre B, with the chord of 60°, describe an arc, and with the chord of the number of degrees that the angle is to contain, cut off the portion required, and draw the line BD through its extremity. PROBLEM XVIII. To inscribe a circle in a given triangle ABC. and C, by the lines Ao and Co. PROBLEM XIX. n B In a given circle, to inscribe an equilateral triangle, a hexagon, or a dodecagon. From any point A in the circumference, as a centre, with a distance equal to the radius Ao, describe the arc BoF. F Then BF laid three times round the E B COR. The side of a regular hexagon is equal to the radius of the circle in which it is inscribed. PROBLEM XX. To inscribe a square in a given circle. Draw two diameters, AC and DB, to each other, and join AB, BC, CD, and DA; then ABCD will be the square required. Cor. A circle may be described about a given square ABCD, by drawing the diagonals, AB, CD, and their point of intersection, O, will be the centre; from which, with the radius OA, describe a circle, and it will be the circle required. PROBLEM XXI. To describe a square about a given circle. Draw any two diameters, AB, CD, at rs to one another; then through the points C and D draw FG, EH || to AB, or to CD, and through the points A and B draw FE and GH || to CD, or to AB; and the figure FEHG will be a square described about the circle. PROBLEM XXII. E D Ф B B To inscribe a regular polygon of any proposed number of sides in a given circle. Divide 360° by the number of sides of the polygon, and make the LAoB at the centre, containing the number of degrees in the quotient. Then if the points A, B, be joined, and the chord AB applied to the circumference the proposed number of times, it will form the polygon required. PROBLEM XXIII. On a given line AB, to form a regular polygon of any proposed number of sides. Divide 180° by the number of sides of the figure, and subtract the quotient from 90°. Make the LSOAB and OBA each equal to the difference so found; and from the point of intersection O, with the radius OA or OB, describe a circle, and the chord AB being applied to the circumference the proposed number of times, will form the polygon required. PROBLEM XXIV. To make a triangle equal to a given trapezium ABCD. Draw the diagonal DB, and through C draw CE || to DB, meeting AB, produced in E, and join DE; the AADE will be equal to the trapezium ABCD. For the ADEB is the DCB, (Geo., prop. 26.) PROBLEM XXV. To make a triangle equal to any right lined figure ABCDEFG. Produce the side AB both. ways at pleasure, and draw the diagonals EA, EB, and by the last problem make the ΔΕΙΑ= the trapezium AGFE, and the AEHB the trapezium BCDE. Draw IK || to AE, and HL || to EB; then join KE and EL, K and the AKEL will be equal to the figure ABCDEFG. PROBLEM XXVI. A B To describe an oval or figure resembling an ellipse on a given straight line AB. Take any points, C, D, at equal distances from A, B, and on CD describe two isosceles As CED, CFD, and produce the sides to the points p, o, n, m; then from the centres C and D, with the radius AC or DB, describe arcs bounded by the sides of the isosceles As produced; and from the centre F, with radius Fm or Fn, describe the arc mn, and from the centre E, with the radius Ep or Eo, describe the arc po, and the figure thus formed will be an oval. GENERAL EXERCISES IN ALGEBRA. 1. Find the sum of 3a2+4ab+b2, 4a2 — 4ab+b2, —a2+4ab+4b2, 7a2-362, and 5a2+10ab+5b2. Ans. 18a2+14ab+862. 2. Find the sum of 16ac+16c2+4a2, 12c2+7ac+a2, 6a2+4c2, 13ac+144a2+c2, 11a2+22ac+4c2, and 3a2+ 12ac+12c2. Ans. 169a2+70ac+49c2. 3. Find the sum of 14a+√ac-5c, 13a-5/ac+8c, 7a-14/ac, 3c+5a, 7c—6√ ac+5a, 13a+14√ac +3c, and 4/ac-6a+4c. Ans. 51a-6/ac+20c. 4. Find the sum of 4a+76-3c, a+4ab-c, 5a-3ab +4c, 3a-ab+c+3d, 4a+7c—4d, and 5d-c+4a-2b. Ans. 21a+56+7c+4d. 5. Find the difference of 15a2+12ab+1362, and 7a2+ 1162. Ans. 8a2+12ab+262. 6. From 17ac+5a2+6c2, take 17ac-5a2+3c2-4d. Ans. 10a2+3c+4d. 7. From the sum of 3a2+2ax+2x2, and 5a2 +7ax—x”, take the sum of 7a2-7ax+x2, and 12ax-3a2-4x2. Ans. 4a2+4ax+4x2. 8. From a2+b2+c2+2ab+2ac+2bc, take 2ab+2ac2bc-a2-b2c2. Ans. 2a2+262+2c2+4bc. 9. Multiply a+b by c-d. Ans. ac+bc-ad-bd. 10. Multiply a-b+c-d by a+b-c-d. Ans. a2-b2-c2+d2-2ad+2bc. 11. Find the product of (x—10)(x+1)(x+4). Ans. x35x2—46x-40. 12. Find the product of a2+ac+c2, and a2—ac+c2. Ans. a+ac+c. 13. Find the continued product of a+1, a+2, a+3, and a +4. Ans. a1+10a3+35a2+50a +24. 14. Find the continued product of a-x, a+x, a2+ax +x, and a2-ax+x2. Ans. a6-x6. 15. Multiply x2+10xy+7, by x2-6xy+4. Ans. x+4y-60x2y2+11x2-2xy+28. 16. Divide 6x-3x2+12x2+14x, by 3x. 17. Divide x1-81, by x—3. 18. Divide a3+b3, by a+b. Ans. 22-2+4x+48. 19. Divide +y3, by x+y. Ans. x-x3y+x2y2—xy3+y1. 20. Divide x+x2y2+x3y3+y, by x+y3. Ans. 3+y2. 21. Divide x3-px1+qx3-qx2+px-1, by x-1. Ans. (p-1)x3+(q-p+1)x2-(p-1)x+1. 77x3 43.x2 33.x 2 23. Find the sum of the products (a+b+c)(a+b—c), (a+b+c)(a−b+c), and (a+b+c)(—a+b+c). Ans. a2+b+c2+2ab+2ac+2bc. 24. Find the sum of the products (a+x)(a+x), (a—x) (a-x), and (a+x)(a—x). Ans. 3a2+x2. 25. Find the sum and difference of (a+x) (a+x), and (a—x) × (a—x). Ans. Sum 2a2+2x2, diff. 4ax. 26. Find the sum and difference of (a+b+c)(a+b+c), and (a+b—c)(a+b—c). Ans. Sum 2a2+262 +2c2+4ab, and diff. 4ac+4bc. 27. Find the sum of the products of (a+b+c+d) (a+b+c-d), and (a+b-c+d) (a−b+c+d), and also their difference. Ans. Sum 2a2+2ab+2ac+4bc+ 2ad, and their diff. 2ab+2ac-2ad+2b2+2c2-2d2. 28. How much does the product of (a+b+2c)(a+b +2c), exceed the product of (a-b-2c) (a—b—2c). Ans. 4ab+8ac. Ans. a+a+x. 29. Divide a2x2, by aa—x3. 30. Divide a-b by 3 1 Ans. aa+a3¿ì+a1b3+¿a ̧ 1 1 31 31. Find the greatest common measure of a2+ab—12b2, and a2—5ab+6b2. Ans. a-3b. 32. Find the greatest common measure of 6a2+7ab— 362, and 6a2+11ab+362. Ans. 2a+3b. 33. Find the least common multiple of x3-a3, and x2-a2. Ans. x1+xa-xa3—a1. 34. Find the least common multiple of a-1, a2-1, a-2, and a2-4. Ans. at-5a2+4. 35. Find the least common multiple of 2a-1, 4a2-1, and 4a2+1. |