(2) A circle is a plane figure bounded by one line, called the circumference, which is such that all points of it are equally distant from a point within the circle called the centre. 2. “ Draw a straight line perpendicular to a given straight line from a given point in it." Euclid i. 11. Tate, Art. 23. 3. “The angles at the base of an isosceles triangle are equal to each other; and if the equal sides be produced the angles on the other side of the base shall be equal." Euclid i. 5. Tate, Art. 18. SECTION II. 1. “Prove that the sum of the three angles of a triangle equals two right angles. Given the value of two angles of a triangle, how is the value of the third ascertained ?" Euclid i. 32. Tate, Art. 34. The sum of the three angles being two right angles, or 180°; if two be known, they must be deducted from 180°, to leave the third. 2. Prove that parallelograms on the same base and between the same parallels are equal to one another.”' Euclid i. 35. Tate, Art. 48. 3. “ In any right-angled triangle the square which is described upon the side subtending the right angle is equal to the square described upon the sides containing the right angle." Euclid i. 47. Tate, Art. 45. 66 SECTION III. 1. “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." Euclid ii. 5. 2. “ In any triangle the square of the side subtending any acute angle is less than the squares of the sides containing the angle by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall upon it from the opposite angle and the acute angle ;--prove only the first case of this proposition." Euclid ii. 13. Tate, Art. 50. SECTION IV. 1. Construct a triangle whose area shall be equal to that of a given trapezium." From Euclid ii. 14, describe a square equal to the given trapezium. A triangle with a base equal to that of the square so found, and with twice the altitude of the square; or with an altitude equal to that of the square, but a base twice as large will be equal to the square (Euclid i. 42), and consequently to the trapezium. 2. “Show how to make a square double a given square.' By constructing a square on the diagonal of the given square. 3. “Show that the diagonals of a parallelogram bisect each other.' Playfair's Euclid ii. Prop. B. HIGHER BRANCHES OF MATHEMATICS. SECTION J. = 40. 1. Find x from the equation x2 12x = - 35. A draper bought a piece of silk for £16 4s., and the number of shillings which he paid per yard was the number of yards. How much did he buy? 2. Find x and y from the equations x - y = 4, 2 + ya There is a rectangular field whose length exceeds its breadth by 16 yards, and it contains 960 yards : find its dimensions. 3. A fast passenger train starts at 20 minutes past 12; it overtakes a luggage train which travels 15 miles an hour, and after having gone 15 miles further overtakes a slow passenger train which travels 20 miles an hour. Another fast train, which travels at the same rate as the first, starts from the same station at 2 o'cloek of that day, overtakes the luggage train, and, after having gone 65 miles further overtakes the slow train also, and finds that it has then travelled 120 miles. Required the rate at which the fast trains travel. SECTION II. 1. Prove the formula for finding the sum of an arith metical series. The first term of an arithmetical progression is 1; the common difference, 1; the sum of the series, 36. Required, the number of terms. 2. When are quantities said to be proportional ? And when is one quantity said to vary as another ? Prove that, if a : 6:: c:d; then a: a - b::c:c- d. 3. Find the number of different combinations that may be made of n different things, taking r of them together. SECTION III. 1. In a circle the angle in a semicircle is a right angle, but the angle in a segment greater than a semicircle is less than a right angle, and the angle in a segment less than a semicircle is greater than a right angle. 2. Deseribe a circle in a given triangle. 3. Similar triangles are to each other in the duplicate ratio of their homologous (or corresponding) sides, Prove this, and indicate the steps of the proof of the corresponding proposition for all similar figures. SECTION IV. 1. Define the sine and the tangent of an angle. What are the values of sin. 90°, tan. 180°, and tan. 45°. Prove that cos. 2 A = 2 cos. ?A · 1 =1 2 sin. ?A. 2. Having given one side of a right-angled triangle and the angle adjacent to that side, show by what calculations the other parts of the triangle may be obtained. 3. Write down the expression for the cosine of an angle of a triangle in terms of the sides; and prove the expression for the sine of an angle, and for the area of the triangle, in terms of the sides. What is the logarithm of a number? What are the properties of logarithms on which the utility of logarithmic tables depends? SECTION V. 1. Prove the expressions for the circumference and area of a circle in terms of its radius. 2. Show that the solid content of any cone, or pyramid, is found by multiplying the area of its base by one-third its height. SECTION 1. 1. Find x from the equation x 12 2 = 35....(a) "A draper bought a piece of silk for £16 4s., and the number of shillings which he paid per yard was $ the number of yards. How much did he buy?" (6) (a)....x— 12x = -35; completing the square we get x- 12 x + 36 = 36 — 35 = 1 324 (6) Let x = number of yards; .. = price per yard in shillings. But by the question this is 4 x equal to 9 4 x 324 9 2c2 = 81 x 9 x = 9 X 3 2. (a)" Find x and y from the equations x-y=4, x2 + y2 = 40. = 27 |