88. Describe a square which shall have the arms of one of its angles falling upon the sides of a given rightangled triangle and the vertex of the opposite angle on the hypothenuse. 89. Find a square equal to the difference between two given squares. 90. From AC the diagonal of a square ABCD, cut off AE equal to one-fourth of AC, and join BE, DE. Shew that the figure BADE equals twice the square on AE. 91. Squares are described on the three sides of a rightangled triangle, and the adjacent angular points of the squares joined; prove that each of the triangles so formed equals the given triangle. 92. Squares are described on the sides of a rightangled triangle, and the adjacent angular points of the squares joined. Prove that the area of the hexagon thus formed is equal to the square on the hypothenuse together with the square of a line equal to the sum of the other two sides. 93. If the diagonals of a quadrilateral cut each other at right angles, prove that the squares described on two opposite sides will be equal to the squares on the other two sides. 94. If ACB be any triangle, and CD be drawn 1 to AB, prove that the difference between the squares on AD, DB is equal to the difference between the squares on AC, CB. BOOK II. 1. The sum of the perpendiculars let fall on the sides of an equilateral figure from any point within it is constant. 2. Divide a straight line into two parts, so that the sum of their squares may be the least possible. 3. If a straight line be drawn from the vertex of an isosceles triangle to any point in the base, the square on this line together with the rectangle under the segments of the base equals the square of either of the other sides. 4. Divide a straight line into two parts so that the rectangle contained by the whole and one of the parts may be equal to the rectangle contained by the other part and a given straight line. 5. Divide a straight line into two parts so that the rectangle contained by the segments shall be the greatest possible. 6. Divide a straight line into two parts so that the difference of the squares on the parts is equal to twice the rectangle contained by the parts. 7. If from one of the equal angles of an isosceles triangle, a perpendicular be drawn to the opposite side, the rectangle contained by that side and the segment of it intercepted between the perpendicular and base is equal to half the square on the base. 8. If ABC be an isosceles triangle, and DE be drawn parallel to the base BC and EB joined : prove that square on BE = rect. (BC, DE) + square on CE. 9. Divide a straight line into two parts so that the squares on the whole line and on one of the parts shall be together equal to three times the square on the other part. 10. Prove that the squares on the diagonals of a trapezoid are together equal to the squares on its two sides which are not parallel and twice the rectangle contained by the sides which are parallel. 11. Divide a straight line into two parts so that the rectangle contained by them may be equal to a given rectangle. 12. Describe a rectangle equal to a given square, and having its perimeter equal to a given straight line. 13. In any triangle ABC, if BP, CQ be drawn respectively perpendicular to CA and BA, produced if necessary ; shew that square on BC = rect. (BA, BQ) + rect. (CA, CP). 14. The squares on two sides of a triangle are together double of the squares on the straight line drawn from the vertex to the middle point of the base and on half the base. 15. Given the lengths of the three lines drawn from the angles of a triangle to the points of bisection of the opposite sides, construct the triangle. 16. Describe a rectangle equal to a given square, and also having the difference between its adjacent sides equal to a given straight line. 17. If a perpendicular be let fall from the vertex of a triangle upon the base, then will the difference between the squares on the sides be equal to the rectangle contained by the base and the difference between the segments of the base. 18. The square on the hypothenuse of a right-angled triangle is < the square on a straight line which is = the sum of its sides by twice the rect. contained by those sides. Hence deduce the theorem of Pythagoras. 19. The square on the hypothenuse of a right-angled triangle is > the square on a straight line which is = the difference of its sides by twice the rectangle contained by those sides. 20. The square on the sum of the sides of a rightangled triangle together with the square on their difference is double of the square on the hypothenuse. N.B. Exercises (18), (19), (20) may be easily established by means of the accompanying figure. 22. The squares on the lines drawn from any given point to the extremities of one diagonal of a rectangle are together equal to the squares on the lines drawn to the extremities of the other diagonal. 23. The base of a triangle is fixed, and the sum of the squares on the sides is constant; determine the locus of the vertex. 24. In every parallelogram the squares on the sides are together equal to the squares on the diagonals. 25. What triangles are those whose sides are (13, 4, 12) (13, 5, 12) (13, 6, 12) (13, 6, 7) (13, 6, 6)? 26. What triangles are those whose sides are (12, 10, 8) (10, 8, 6) (8, 6, 4) (6, 4, 2)? 27. Produce a straight line so that the rectangle contained by the whole .line and the part produced shall be equal to the square on the given straight line. 28. Produce a straight line so that the rectangle contained by the given line and the part produced may be equal to a given triangle. 29. Draw two squares equal to a given square : one of them being double of the other. 30. Draw a square which shall be equal to two given triangles together. 31. The perimeter of an equilateral triangle is greater than that of an equal square. 32. Produce a given straight line so that the square on the whole line thus produced may be double the square on the part produced. 33. Of all equal rectangles the square has the least perimeter. |