4. Inscribe a regular hexagon in an equilateral triangle. It is clear that each side must be trisected. H 5. Inscribe a rhombus in a parallelogram with a given point Examine the cases in which either two or four of the vertices of the rhombus are on the produced sides. 6. Construct a trapezium whose sides are given. E A Construct the AEAB so that EB, BA are the oblique sides and EA the difference between the parallel sides. Complete the EDCB. ABCD is the trapezium required. H A 7. Construct a quadrilateral having given two opposite sides and three angles. Let AB, CD=the given sides, and 4s A, B, C=the given angles. Draw AB, make ▲ BAD= ▲ A, and Cut off BECD. Draw ED | BC meeting AD in D. Draw DC || EB meeting BC in C. Prove ABCD to be the required quadrilateral. 8. Divide a given straight line into two parts, so that the sum of the squares on the segments may be equal to the square on a given line. A F E B Let AB be the given line to be divided, AC2 the given square. If ABC= rt. 2, EC=EB, AE2+ EB2=AE2+ EC2=AC2. Also AF2+FB2=AF2+FD2=AD2. Hence we obtain the construction indicated in the figure. 9. Divide a given straight line into two parts so that the difference of the squares on the segments may be equal to the square on a given straight line. Let AB be the given straight line, CD2 the given square. on base CD construct right-angled ACDE, 10. Produce a given straight line, so that the difference of the squares on the whole line thus produced, and on the part produced, may be equal to the square on a given straight line. 44 Let AB be the given straight line, CD2 the given square. Hence derive a construction to obtain ▲ APB and to find Q. 11. Divide a straight line AB in Q so that AQ2=2BQ2. If APQ be a rt. 4, and AP=PQ=QB, 12. Produce a straight line AB to Q so that AQ2=2BQ2. If AQ2=PQ2=PB2+BQ2=2BQ2, we see that Q=1; rt. ≤ ; .. Art. 2. [Euc. I. 32. Thus we obtain BP, and therefore BQ. P A B 13. Construct a square, having given the sum of a side and a diagonal. Given AB, make ABP-1 rt. 4. Draw AP LAB, and take AQ=AP. (AB may also be divided as in Ex. 11.) 14. Draw a straight line BD such that BD2=3AB2. This problem may be solved as in § 22, Exx. 6 and 21, or by the following method: A Draw BDLAB. If BD2=3AB2, ... if AB be produced to C so that 15. Divide a straight line AB in Q so that AQ2=3BQ2. B If PQ=QB, AP2=4PQ2. Thus PAQ half the angle of an equilateral triangle, and PBQ=half a right angle. Hence obtain a construction. 16. Having given two circles, draw a line equal and parallel to a given straight line with one of its extremities on each of the circles. Draw AQ | CP, and show that QP is the required line. Show that there are in general two solutions, and find in what case there is no solution. 17. Inscribe a rhombus in a given triangle so that one angle of the rhombus may coincide with one angle of the triangle. Compare Ex. 3. 18. Inscribe a square in a right-angled triangle so that one angle of the square may coincide with the right angle. Use the method of Ex. 3. 19. Inscribe a square in a given rhombus. Draw the diagonals and compare previous Ex. 20. Divide a straight line AB in C so that AC2=4BC2. Trisect the line. See § 22, Ex. 9. 21. Divide AB in C so that AC2=}AB2. See § 20, Ex. 2. 22. Produce AB to C so that AC2=2AB2. See § 20, Ex. 2. 23. Construct a square, having given the difference between a side and a diagonal. Compare Ex. 12. 24. Produce a straight line AB to C so that AC2=3BC2. Use the method of Ex. 15, but bisect the exterior right angle at B. 25. Find two straight lines, having given the sum of the lines and the difference of their squares. Compare Ex. 9. 26. Find two straight lines, having given the difference of the lines and the difference of their squares. Compare Ex. 10. 27. Given two circles and a point, draw a line through the point to meet the two circles and be bisected at the point. Use § 15, Ex. 1, and a construction similar to that of Ex. 16. Find the condition that this problem may have two, one, or no solutions. |