EASY EXERCISES ON THE FIRST FOUR BOOKS. 1. On a given straight line to describe an isosceles triangle that shall have each of its equal sides double of the base. 2. The line which bisects the vertical angle of an isosceles triangle, bisects the base perpendicularly. 3. To describe an isosceles triangle, the base and one of the sides being given. 4. The difference between two sides of a triangle is less than the third side. 5. To trisect a right angle, that is, to divide it into three equal parts. 6. To trisect a given finite straight line. 7. If the vertical angle of an isosceles triangle be a right angle, each of the angles at the base is half a right angle. 8. In the diagram Book I. Prop. 5. draw AH to the point H in which CF and BG intersect: show that CH is equal to BH; FH to GH; and that the angle BAC is bisected by AH. 9. The angles of a quadrilateral are equal to four right angles. 10. To describe a square equal to the difference of two given squares 11. A line joining the middle points of two sides of a triangle is parallel to the base, and equal to half of it. 12. To bisect a given parallelogram by a line drawn from a point in one of its sides. 13. To bisect a given triangle by a line drawn from a point in one of its sides. 14. To divide a given straight line into two parts, so that their rectangle may be equal to a given square. 15. To produce a given straight line, so that the rectangle of the whole line and the given line may be equal to a given square. 16. The square on the base of an isosceles triangle, whose vertical angle is a right angle, is equal to four times the area of the triangle. 17. The squares of the diagonals of a parallelogram are equal to the sum of the squares of the four sides. 18. With a given radius to describe a circle which shall pass through two given points. 19. The arcs intercepted between any two parallel chords in a circle are equal. 20. From one extremity of a line which cannot be produced, to draw a line perpendicular to it. 21. Given the vertical angle, the base, and the altitude of a triangle, to construct it. 22. To divide a circle into two parts, such that the angle contained in one segment shall equal twice the angle contained in the other. 23. If the diameter of a circle be one of the equal sides of an isosceles triangle, the base will be bisected by the circumference. 24. To describe a circle which shall pass through a given point, and which shall touch a given straight line in a given point. 25. To describe a circle which shall touch a given straight line, and pass through two given points. 26. To trisect a given circle by dividing it into three equal sectors. 27. To describe an equilateral triangle about a square. 28. To draw a tangent to a given circle parallel to a given straight line. 29. To draw a tangent to a given circle making a given angle with a given straight line. 30. Find algebraical formulæ corresponding to the enunciations of the Second Book. 31. A flag-staff of a given height is erected on a tower, the height of which is also given; at what point on the horizontal line drawn from the foot of the tower will the flag-staff appear under the greatest angle? THE END. C. and J. Adlard, Printers, Bartholomew Close. |