EXAMPLES LII. 1. Describe a parallelogram equal in area to a given parallelogram having an angle equal to a given angle. 2. Describe a triangle equal in area to a given triangle having an angle equal to a given angle. 3. Describe a triangle equal in area to a given parallelogram having an angle equal to a given angle. 4. Describe a rectangle equal in area to a given triangle. 5. Describe a right-angled triangle equal in area to a given parallelogram. 6. Describe a rhombus equal in area to a given parallelogram having one of its sides equal to the longer side of the parallelogram. Proposition 44. 149. To describe a parallelogram equal in area to a given triangle having one of its sides equal to a given finite straight line, and one of its angles equal to a given angle. Let C be the given triangle, AB the given finite straight line, and let D be the given angle; it is required to describe a parallelogram equal in area to C, having one side equal to AB, and one angle equal to D. Construction. Describe the parallelogram AEFG, and let this parallelogram be so placed Through B, draw the line BH parallel to GA meeting FG produced in H. Produce the lines HA, FE, (which are not parallel) to meet in K. [Prop. 31] Draw the line KML through K, parallel to FG, or EA, to meet GA, HB respectively in M, L. [Prop. 31] Proof. Because FKLH is, by construction, a parallelogram and FA, AL are the complements of the parallelograms GB, EM which are about its diameter HK, therefore the complements FA, AL are equal in area. [Prop. 43] Hence, ABLM, which is by construction a parallelogram, is equal in area to FA, and therefore to the triangle C ; also one of its sides is AB, and one of its angles BAM is equal to GAE and therefore to D. Wherefore, a parallelogram has been described equal in area, etc. Q.E.F. EXAMPLES LIII. 1. Construct a rectangle equal in area to a given triangle, having one of its sides equal to a given finite straight line. 2. Construct a rectangle equal in area to a given square, having one of its sides equal to a given finite straight line. 3. Construct a triangle equal in area to a given triangle, having one of its sides equal to a given finite straight line and one of its angles equal to a given angle. 4. Construct a right-angled triangle equal in area to a given square, having one of its sides containing the right angle equal to a given finite straight line. 5. On one side of a given triangle construct a rectangle equal in area to the triangle. 6. Construct a rectangle equal in area to the difference of two given squares. L. E. 11 Proposition C. 150. To describe a triangle equal to a given rectilinear figure. Let ABCDE be the given rectilinear figure; it is required to describe a triangle equal in area to ABCDE. Construction. Join B, E, the two angular points adjacent to A. Through A draw a line AH parallel to BE. [Prop. 31] let H be the point in which AH meets CB. Then EHCD is a rectilinear figure, equal in area to ABCDE, and having one side fewer than ABCDE. Proof. Because the triangles ABE, HBE are upon the same base CB, and between the same parallels AH, BE; therefore the triangles ABE, HBE are equal in area ; [Prop. 37] so that the whole area HCDE is equal to the area ABCDE. Again, by repeating the above process with respect to the angular point H, as in Figure (ii), we obtain a triangle equal in area to the four-sided figure HCDE, and therefore to ABCDE. Wherefore, a triangle has been described, etc. Q.E.F. EXAMPLES LIV. 1. A given four-sided figure has one of its angles a right angle; construct a right-angled triangle equal to it in area. 2. On a given finite straight line as diagonal construct a rhombus equal to a given parallelogram. 3. To a given straight line apply a rectangle equal in area to a given rectilineal figure of four sides. |