918 If a line bisects the bases of a trapezoid, it bisects the trapezoid. 919 If similar polygons are constructed on the sides of a right triangle as homologous sides, the polygon on the hypotenuse is equivalent to the sum of the polygons on the legs. 920 If P is any point in the circumference of a circle whose diameter is AB, the sum of the squares on PA and PB is constant. 921 If two equal circles so intersect that the circumference of each passes through the center of the other, the square on their common chord is equivalent to three times the square on the radius. 922 The sum of the squares of the sides of a parallelogram is equal to the sum of the squares of the diagonals. 923 If the mid-points of two adjacent sides of a parallelogram are joined, the triangle thus formed is one eighth of the parallelogram. 924 If AB, BC, and CA of the triangle ABC are each produced its own length to B', C', and A' respectively, the triangle A'B'C' is seven times the triangle ABC. B 925 If the sides of a parallelogram ABCD are produced in order, each its own length, to A', B', C', D', A'B'C'D' is a parallelogram five times the parallelogram ABCD. 926 The square inscribed in a sector which is one fourth of a circle is five eighths of the square inscribed in the semicircle. [Let q be a side of the square in the sector and s a side of the square in the semicircle whose radius is 927 The square inscribed in a semicircle is two fifths of the square inscribed in the circle. 928 If two equivalent triangles have an angle in each equal, the including sides are inversely proportional. 929 If from the mid-point of the base of a triangle lines are drawn parallel to the other two sides, the parallelogram thus formed is equivalent to half the triangle. NOTE. Exercises 930-939 relate to the annexed figure, repeated from § 418. 930 The points E, B, F, are in a straight line. 931 AD and CG are parallel. 932 CE and BK are perpendicular lines. 933 BC produced bisects FH at Q, and CQ equals the half of AB. 934 The sum of the perpendiculars dropped from E and F to AC produced is equal to AC. 935 The sum of the angles HCF and KAE is equal to three right angles. 936 If DG, EK, and FH are joined, each of the triangles DBG, EAK, and FCH is equivalent to the triangle ABC. 937 BA2 + BH2 = BC2 + BK2. 938 FH2 = AB2 + 4 BC2. 939 FH2 + EK2 = 5 AC2. 940 If two equivalent triangles are on opposite sides of the same base, the common base bisects the line joining their vertices. 941 If the medians of a triangle ABC intersect in O, the triangle BOC is one-third the triangle ABC. 942 Four times the sum of the squares of the medians of any triangle is equivalent to three times the sum of the squares of the sides. 943 In a triangle whose base is b and altitude a, prove that the side of ab the inscribed square is equal to a + b 944 In the annexed figure, ABCD is a square; E, F, A G, and H are the mid-points of the sides. Prove IJKL a square equivalent to one fifth of the square ABCD. E 945 The sum of the squares of the four sides of a quadrilateral is equal to the sum of the squares of the diagonals plus four times the square of the line joining B the mid-points of the diagonals. H F L K 946 In a quadrilateral the lines which join the mid-points of the opposite sides and the line which joins the mid-points of the diagonals meet in a point of bisection. PROPOSITION XI. PROBLEM 423 To construct a square equivalent to the sum of two given squares. H DATA. H and K are two given squares. REQUIRED. To construct a square equivalent to H+K. SOLUTION Construct the rt. A ABC whose legs are respectively equal to the sides of the squares H and K. Construct the square R whose side is equal to the hypotenuse AC. "The square on the hypotenuse of a rt. A is equivalent to the sum of the squares on the other two sides." .. R ≈ H + K. § 418 Q. E. D. EXERCISES 947 Construct a square equivalent to the sum of two squares whose sides are 3 and 4. 948 Construct a square equivalent to the sum of two squares whose sides are 8 and 15. 949 Compute the side of a square whose area is equal to the areas of two squares whose sides are 13 in. and 84 in. 950 Compute the side of a square whose area is equal to the areas of two squares whose sides are 20 in. and 99 in. PROPOSITION XII. PROBLEM 424 To construct a square equivalent to the difference of two given squares. K H DATA. H and K are two squares, K being the larger. Ꭱ SOLUTION Construct the rt. ▲ ABC in which the leg AB is equal to the side of H, and the hypotenuse AC is equal to the side of K. Construct the square R whose side is equal to BC. "The square on either leg of a right triangle is equivalent to the difference of the squares on the hypotenuse and on the other leg." .. RK — H. § 419 Q. E. D. EXERCISES 951 Construct a square equivalent to the difference of two squares whose sides are 12 and 13. 952 Construct a square equivalent to the difference of two squares whose sides are 24 and 25. 953 Compute the side of a square whose area is equal to the difference of the areas of two squares whose sides are 35 and 37. 954 Compute the side of a square whose area is equal to the difference of the areas of two squares whose sides are 91 and 109. PROPOSITION XIII. PROBLEM 425 To construct a square equivalent to the sum of any number of given squares. b a E d DATA. a, b, c, d, are sides of given squares. REQUIRED. To construct a square equivalent to the sum of the squares on a, b, c, d. SOLUTION Construct the rt. ▲ ABC with legs a and b. 426 SCHOLIUM. By this problem we can construct a line equal to the square root of any number. Thus, if a=b=c= 1, AD = √3. Compute the side of a square equivalent to the sum of the squares whose sides are: 956 12, 21, 28. 957 7, 24, 60, 72. 958 16, 25, 33, 39, 52, 56, 60, 63. 959 13, 36, 40, 51, 68, 75, 77, 84. NOTE. The answer to each of the last four exercises is a whole number. |