EUCLID, BOOK II. CONTENTS. 16 Definitions and General Questions. Enunciate the proposition which proves that the area of a triangle is represented by half the rectangle which has the same base and altitude. Explain the term gnomon. Questions on the Propositions. 1. What is a rectangle, and how is it said to be contained ? Prove that if there be two straight lines, one of which is divided 2. Prove that if a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts are together equal to the square on the whole line. Deduce from this that a square on a straight line is equal to four times 3. If a straight line be divided into any two parts, the rect- angle contained by the whole and one of the parts is equal to 16 Same proposition. Divide a given line so that the rectangle contained by the whole line and one part may be twice the square on the other part. 4. Define a rectangle and a square. Prove that if a straight line be divided into any two parts, the square on the whole linė is equal to the squares of the two parts, together with twice the rectangle contained by the parts. The parallelograms about the diagonal of a square are also squares. Same proposition. If a straight line AB be divided in C, the squares on AC, CB together are least when C is the middle point of AB. 5. If a straight line be divided into two equal parts, and also into two unequal parts, the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line. The difference between the squares on any two straight lines is equal to the rectangle contained by the sum and difference of those lines. Prove it. 6. If a straight line be bisected and produced to any point, the rectangle contained by the whole line thus produced, and the part of it produced together with the square on half the line bisected, is equal to the square on the straight line which is made up of the half and the part produced. 7. Define a rectangle and a gnomon. Prove that if a straight line be divided into any two parts, the squares of the whole line and of one of the parts are together equal to twice the rectangle contained by the whole and that part together with the square on the other part. Same proposition. Prove this both geometrically and algebraically. Same proposition. Show that the sum of the squares on two lines is never less than twice their rectangle. 8. If a straight line be divided into any two parts, four times the rectangle contained by the whole line and one of the parts, together with the square of the other part, is equal to the square of the straight line, which is made up of the whole and that part. 9. If a straight line be divided into two equal and also two unequal parts; the squares on the two unequal parts are together double of the square on half the line, and of the square on the line between the points of section. Same proposition. Divide a straight line into two parts so that the sum of their squares may be the least possible. 10. If a straight line be bisected and produced to any point, the square on the whole line thus produced, and the square on the part of it produced, are together double of the square on half the line bisected, and of the square on the line made up of the half and the part produced. 11. Show how to divide a given line into two parts so that the rectangle contained by the whole and one of the parts shall be equal to the square of the other part. : Same proposition. Show that the squares on the whole line and one part are equal to three times the square on the other part. 12. In an obtuse-angled triangle, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle, by twice the rectangle contained by the side upon which, when produced, the perpendicular falls, and the line without the triangle between the perpendicular and the obtuse angle. ABDE is a square on AB, the hypotenuse of a right-angled triangle ABC Show that the difference of the squares on CD, CE is equal to the difference of the squares on CB, CA. Enunciate Euclid, Prob. 12, Book II. If the obtuse angle be two-thirds of two right angles, show that the square of the opposite side is equal to the squares of the containing sides, together with the rectangle of those sides.. 13. By what do the squares on two sides of an acute-angled triangle exceed the square on the third side ? Prove the truth your answer: Ín every triangle the square on the side subtending an acute angle is less than the squares on the sides containing that angle by twice the rectangle contained by either of those sides, and the straight line intercepted between the perpendicular let fall on it from the opposite angle and the acute angle, of 14. Show how to describe a square that shall be equal to a given rectilineal figure. Deductions. 1. If a straight line be divided into two equal, and also into two unequal parts, the part between the points of section is equal to half the difference of the unequal parts. 2. If the sides of a triangle be as 2, 4, 5, show whether it will be acute or obtuse angled. 3. Divide a given straight line into two parts, so that the square on one part shall be double the square on the other. 4. In any triangle BAC, a line AD is drawn, bisecting BC in D. Show that the sum of the squares on AB, AC is equal to twice the sum of the squares on AD, BD. 5. Show that if a straight line be divided into two parts, so that the rectangle contained by the whole and the first part is equal to the square of the second part, then must the squares on the whole line and on the first part be equal to three times the square on the second part. 6. From AC, the diagonal of a square ABCD, cut off AE equal to one-fourth of AC, and join BE, DE. Show that the figure BADE equals twice the square on AE. 7. Draw DE parallel to BC, the base of an isosceles triangle ABC, and join BC. Show that the square on BE is equal to the square on CE, together with the rectangle BC, DE. (Hints :-Draw EF parallel to AB, and EG perpendicular to BC. Then 1. 47, II. 5.) 8. Enunciate (only) Props. 12 and 13 of Second Book. The sum of the squares on the sides of a parallelogram is equal to the sum of the squares on the diagonals. 9. If the square on the perpendicular from the vertex of a triangle to the base is equal to the rectangle contained by the segments of the base, the vertical angle is a right angle. (Apply 1. 47, II. 4.) 10. If from the middle point of one of the sides of a rightangled triangle a perpendicular be drawn to the hypotenuse, the difference of the squares on the segments so formed is equal to the square on the other side. 11. If A be the vertex of an isosceles triangle ABC, and CD be drawn perpendicular to AB, prove that the squares on the three sides are together equal to the square on BD, twice the square on AD, and thrice the square on CD. 12. Enunciate Prop. 13 of the Second Book, and prove that the sum of the squares on any two sides of a triangle is equal to twice the sum of the squares on half the base, and of the line joining the vertical angle with the middle point of the base. 13. If from any point perpendiculars be drawn to all the sides of any rectilineal figure, the sum of the squares on the alternate segments of the sides will be equal. 14. If from the extremities of any chord in a circle lines be drawn to any point in the diameter to which it is parallel, the sum of their squares is equal to the sum of the squares of the segments of the diameter. Miscellaneous. 1. The centre of a circle being given, find two opposite points in the given circumference by means of a pair of compasses only. 2. Write out the algebraical results corresponding to the enunciations of Propositions 1, 4, 7, 8, Book II. |