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No. 36. — Euclid. — For Completion of Class D, under Regulations now repealed. Time allowed : Three hours. 1. If two triangles have the three sides of the one equal to the three sides of the other, the triangles are equal in all respects. If A B C D is a rhombus, and A C cuts B D in E, then A E is equal to E C. 2. Bisect a given rectilineal angle. Compare Euclid's method with any practical one with which you are acquainted. 3. If one side of a triangle be produced, the exterior angle is greater than either of the interior and opposite angles. Two of the medians of an isosceles triangle are equal. 4. If a straight line fall on two parallel straight lines, it makes the alternate angles equal, and the exterior angle equal to the interior and opposite angle on the same side; and also the two interior angles on the same side together equal to two right angles. If any straight line be parallel to the line joining two points, it is equidistant from those points. 5. Triangles on equal bases and between the same parallels are equal to one another. Make a triangle equal in area to a given quadrilateral A B C D. 6. In any right-angled triangle, the square on the side subtending the right angle is equal to the sum of the squares on the sides containing the right angle. 7. If a straight line be divided into any two parts, the square on the whole line is equal to the sum of the squares on the two parts, together with twice the rectangle contained by the two parts. In a right-angled triangle, if a perpendicular be drawn from the right angle to the hypotenuse, the square on this perpendicular is equal to the rectangle contained by the segments of the hypotenuse. 8. In 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 these sides, and the straight line intercepted between the perpendicular let fall on it from the opposite angle and the acute angle. If from one of the base angles of an isosceles triangle a perpendicular be drawn to the opposite side, then twice the rectangle contained by that side and the segment of it adjacent to the base is equal to the square on the base.

No 37. — Geometry and Trigonometry. — For Class C and for Civil Service Senior. Time allowed : Three hours. 1. Triangles on the same base and between the same parallels are equal in area. Find the triangle that has the least perimeter, with a given base and area. 2. In every triangle the square on the side subtending an acute angle is less than the sum of 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. C is any point on the circumference of a circle whose centre is the middle point of a given straight line AB : prove that the sum of the squares on AC, BC is constant. 3. Equal chords in a circle are equidistant from the centre, and chords that are equidistant from the centre are equal. In a given circle draw a chord that shall be equal to one given straight line and parallel to another. 4. Angles in the same segment of a circle are equal. Two circles intersect at A and B, and through A any straight line PAQ is drawn terminated by the circumferences: prove that PQ subtends a constant angle at B. 5. Inscribe a circle in a given triangle. If the circle inscribed in the triangle ABC touches the sides at D, E, F, prove that the angles of the triangle DEF are respectively 90°-A/2, 90°-B/2, 90°-C/2. 6. If a straight line be drawn parallel to a side of a triangle it shall cut the other sides, or those sides produced, proportionally. From P, a given point in the side AB of the triangle ABC, draw a straight line to AC produced so that it shall be bisected by BC. 7. Give a definition of an angle, suitable for the purposes of trigonometry. Explain the different systems of measuring angles, and the advantages and disadvantages of each system. What is meant by positive and negative angles ? What is the circular measure of the angle subtended at the centre by an arc of length 2-7 in. if the radius of the circle is 5 in. ? 8. Prove the formulae— Sin (A+B) = sinA cosß + cosA sinß Cos (A+B) = cosA cosß — sinA sinß and deduce expressions for sin 2 A and sin3A in terms of sinA and cos A. 9. Prove the identities— (i.) sin 8 A + cos 2 A = 1 (■»■) * + r+iisr = sec A (iii.) cos (A +B) cos (A-B) = cos a B-sin a A (iv.) cos 4 A —sin 4 A = cos 2A