PROBLEM XI. 311. To construct a rectangle, given its area and the sum of the base and altitude. Let AB be equal to the given sum of the base and altitude, and let the given area equal that of the square whose side is a. On AB as a diameter, describe a semicircle, and at A erect the LAC - a. Draw CD to AB, cutting the circumference at D, and from D draw DPL to AB. Now, DP2 = : AP × PB = a2; (282) ... AP is the base and PB is the altitude of the required rectangle. And APFE, whose altitude PF PB, is the required rectangle. Q. E. F. PROBLEM XII. 312. To construct a rectangle, given its area and the difference of the base and altitude. Let AB equal the given difference of the base and altitude, and let the given area be that of the square whose side is a. C P α On AB as a diameter, describe a O. At A draw the tangent AC a, and draw the secant .'. CD is the base and CP is the altitude of the rectangle required, and the rectangle can readily be constructed. Q. E. F. PROBLEM XIII. 313. To construct a square having a given ratio to a given square. Let op be the given ratio, and S the given square whose side is a. o, and BC = p. On the indefinite line AM, cut off AB On AC as a diameter, describe a semicircle. At B erect a cutting the circumference at D, and draw AD and CD. Take DF = a, and draw EF to AC. Now, DE: DF :: DA : DC (267), which gives ᎠᎬ : ᎠᎰ :: ᎠᎪ : ᎠᏅ (170) ... the square whose side is DE is the required one. Q.E.F. 314. SCHOLIUM.-Since similar polygons are to each other as the squares of their homologous sides, we can find, by means of the above problem, the homologous side of a polygon similar and having a given ratio to a given polygon. EXERCISES IN INVENTION. the THEOREMS. 1. The square inscribed in a circle is equivalent to half square described on the diameter. 2. Prove geometrically that the square described on the sum of two lines is equivalent to the sum of the squares described on the two lines plus twice the rectangle of the lines. 3. Prove geometrically that the square described on the difference of two lines is equivalent to the sum of the squares described on the two lines minus twice the rectangle of the lines. 4. Prove geometrically that the rectangle of the sum and difference of two lines is equivalent to the difference of the squares described on the lines. 5. If a straight line is drawn from the vertex of an isosceles triangle to any point in the base, the square of this line is equivalent to the rectangle of the segments of the base together with the square of either of the equal sides. 6. The area of a circumscribed polygon equals half the product of the perimeter and the radius of the inscribed circle. 7. The triangle formed by drawing straight lines from the extremities of one of the non-parallel sides of a trapezoid to the middle point of the other, is equivalent to half the trapezoid. 8. The two triangles formed by drawing straight lines from any point within a parallelogram to the extremities of either pair of opposite sides, are equivalent to half the parallelogram. 9. The bisector of the vertical angle of a triangle divides the base into parts proportional to the adjacent sides of the triangle. PROBLEMS. 1. Trisect a given straight line. 2. Bisect a parallelogram by a line passing through any given point in the perimeter. 3. Construct a parallelogram whose surface and perimeter are respectively equal to the surface and perimeter of a given triangle. 4. On a given straight line construct a rectangle equivalent to a given rectangle. 5. Construct a polygon similar to a given polygon and whose area is in a given ratio to that of the given polygon. |