7. Find sin 18°.

If cos A=tan B and cos B=tan A;
Prove that cos A=2 sin 18°.

8. Prove the formula

a2=b2+c2-2bc cos A,

and apply it to prove that if the straight line which bisects the vertical angle of a triangle also bisects the base, then the triangle must be isosceles.

9. Find the area of a triangle in terms of the sides.

10. Find the radius of the circle which touches one side of a triangle and the two other sides produced.

If ABC and DEF be two triangles with the same perimeter, prove that the radius of the escribed circle touching the side BC of the first is to the radius of the described circle touching the side EF of the second as D

A

tan

is to tan

2

11. Given two sides b and c, and the included angle A of a triangle, find a formula for determining the side a, without the previous determination of the angles B and C, adapted to logarithmic computation.

Apply to the case of b=354 yds., c=426 yds., A=49° 16'.

12. A building on a square base ABCD has two of its sides AB and CD parallel to the bank of a river.

An observer standing on the river's bank in the same straight line with DA finds that the side AB subtends, at his eye, an angle of 45°.

Having walked a yards along the bank he finds that the side DA subtends an angle whose sine is . Prove that the length of each side of the building is yards.

a

V2

XCIX.

1. Construct a triangle of which the sides shall be equal to three given straight lines, but any two whatever of these must be greater than the third.

How would the process fail if the last condition were not fulfilled?

2. Parallelograms on the same base and between the same parallels are equal to each other.

Show that if two triangles have two sides of the one equal to two sides of the other, each to each, and the sum of the two included angles equal to two right angles, the triangles are equal.

3. In a 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.

Show how to construct a straight line, the square on which shall be any given multiple of a given square. 4. If a straight line be bisected and produced to any point, the rectangle contained by the whole line thus produced, and the part 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.

5. The angle at the centre of a circle is double of the angle at the circumference on the same base that is on the

same arc.

6. If two straight lines cut one another within a circle, the rectangle contained by the segments of the one shall be equal to the rectangle contained by the segments of the other.

In a given straight line AB, find a point O such that the rectangle contained by the segments AO and OB shall be equal to a given rectangle not greater than the square on half of AB.

7. Describe a circle about a given triangle.

Lines are drawn parallel to one side of a triangle

cutting the other sides or the other sides produced. Show that if circles be described about the new triangles so formed, their centres all lie in a straight line, and find the angles which this line makes with the two sides of the original triangle.

8. If the vertical angle of a triangle be bisected by a straight line which also cuts the base, the segments of the base shall have the same ratio which the other sides of the triangle have to one another.

If ABC be the triangle, A the vertical angle, BE and CE the segments of the base, and if circles be described about the triangles ABE, ACE, the diameters of these circles are to each other in the same ratio as the segments of the base.

9. If four straight lines be proportionals, the rectangle contained by the extremes shall be equal to the rectangle contained by the means.

On a given straight line describe an isosceles triangle equal to a given triangle.

10. Similar triangles are to each other in the duplicate ratio of their homologous sides.

11. Given a point O in the line AB, find two other points C, D, such that a line OP given in direction shall bisect the angle CPD, and the segments CO, OD, shall bear a given ratio to each other.

C.

ARITHMETIC.

1. What would be the cost of painting the four walls of a room whose length is 24 ft. 8 in., breadth 16 ft. 3 in., and height 11 ft. 6 in., at 4s. a square foot?

2. Reduce to a simple fraction

3. By selling a horse for £82 5s. a person lost 6 per cent.; how much per cent. would he have lost or gained if he had sold the horse for £90 10s.?

4. Find within an inch the diagonal of a square field containing two acres.

5. Multiply, by duodecimals, 10 ft. 5 in. 3 pts. by 9 ft. 7 in. 10 pts. What does the product become when expressed in square inches and a fraction of a square inch?

6. A person invests £2500 in the 9 per cent. Japanese Loan, which was issued at 98 per cent. at 1 discount; how much stock will he have, and what rate of interest will the investment give?

7. Find the difference between the interest and the discount on £2598 15s. for 3 years at 5 per cent. simple interest.

8. Express 4 lbs. 3 oz. 12 dwts. 6 grs. troy weight as a decimal of a pound avoirdupois.

9. Reduce 2671875 to a vulgar fraction, and if the unit be £3 reduce the fraction to shillings, pence, and decimals of a penny.

10. A wall 5 times as high as it is broad, and 8 times as long as it is high, contains 18225 cubic feet. Find the breadth of the wall.

11. A person had £6900 stock paying 3 per cent. in &

company, and was offered the choice of being paid off at par, or of receiving £110 of a new 2 per cent. stock for every £100. He chose the former alternative, and invested his money in the 3 per cent. Consols at 92. Find the amount of his stock in Consols, and the excess of his income above what it would have been if he had agreed to the proposed conversion.

12. Find the prime factors of 111540, 296352, 404352; and thence write down all the numbers which will divide them all without remainder, and the smallest number which they will all divide without remainder.

13. If 56 cubic feet 1044 cubic inches of timber are required to floor a room 29 ft. 3 in. by 25 ft. 4 in., what is the thickness of the boards?

3

14. A tradesman starts with a capital of £960, and after years takes another into partnership with £2100. After 4 years more the whole profits amount to £2304. How ought this to be divided between them?

15. Extract the square root of 2854·7649.

16. Extract the cube root of 1194389981.

17. Multiply, by the method of duodecimals, 6 ft. 7 in. 5 pts. by 8 ft. 3 in. 10 pts.

18. Express the result obtained in the last question in square feet, square inches, and a fraction of a square inch.

19. A tradesman's annual losses during 5 years average 14 per cent. on the capital with which he began, and at the end of the 5 years his effects are worth £2531 5s. What capital did he begin with?

20. A person sells out of the 3 per cent. Consols at 99, and invests in Exchequer Bills, bearing interest at the rate of 21d. a day per cent., when the bills are at a premium of 78. 6d. What effect has this on his income?

21. In the month of December last the number of paupers in a certain union was 336, the number of women being double that of the men, and the children being as