PROBLEM XL. To cut off a Segment from a given Circle, that shall contain a given Angle C. then DEA will be the segment required, any angle E made in it being equal to the given angle C. PROBLEM XLI. To make a Triangle similar to a given Triangle A B.С. NOTE. If ab be equal to A B, the triangles will also be equal, as well as similar. PROBLEM PROBLEM XLII. PROBLEM XLVI. To make a Figure similar to any other given Figure ABCDE. e E c From any angle A draw diagonals b to the other angles. Take Ab a side of the figure required. Then draw bc parallel to BC, and cd to CD, fa and de to DE, &c. : PROBLEM XLIII. To make a Triangle equal to a given Trapezium ABCD. Draw the diagonal DB, and D CE parallel to it, meeting AB produced in E. Join DE; so shall the triangle ADE be equal to the trapezium AB CD. C 10: B PROBLEM NOTE. Nearly in the same manner may a triangle be made equal to any right-lined figure whatever. 4 PROBLEM XLV. To make a Rectangle, or a Parallelogram, equal to a given Triangle ABC. To make a Square equal to a given Rectangle ABCD. Produce one side A B, till BE be equal to the other side BC. Bisect AE in 0; on which as a centre, with radius C F G D Ao, describe a semicircle, and produce BC to meet it at F. On B F make the square BFGH, and it will be equal to the rectangle ABCD, as required. : PROBLEM XLVII. To make a Square equal to two given Squares P and Q. NOTE. Circles, or any other similar figures, are added in the same manner. For if A B and BC be the diameters of two circles, AC will be the diameter of a circle equal to the other two. And if AB and BC be the like sides of any two similar figures, then AC will be the like side of another similar figure equal to the two former, and upon which the third figure may be constructed, by Problem xlii. PROBLEM 15 PROBLEM XLVIII To make a Square equal to the Difference of two given a Squares P, R. On the side AC of the greater square, as a diameter, describe semicircle; in which apply AB the side of the less square. Join BC, and it will be the side of a square equal to the difference between the two P and R, as required, PROBLEM XLIX. To make a Square equal to the sum of any number of Squares taken together. Draw two indefinite lines Am, A n, perpendicular to each other at the point A. On one of these set off AB. the side of one of the given squares, and on the other AC the side of another of them. Join BC, and it will be the side of a square equal to the two together. Then take AD equal to BC, and AE equal to DB n the side of the third given square. So shall DE be the side of a square equal to the sum of the three given squares. And so on continually, always setting more sides of the given squares on the line An, and the sides of the successive sums on the other line A m. NOTE. |