19 two equal chords AB and CD of a circle when produced intersect at a point P provethat PB = PD

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9. A sum of £1,000 is set aside at the end of each year and invested at 5 per cent compound interest. Find approximately the total sum accumulated at the end of 19 years, including the £1,000 set aside'at the end of the nineteenth year. (Use which you require of the following:

(1.05)18=2.406619 (1.05)19=2.526950 (1.05)20=2.653298.) 10. A and B could between them type 4,500 pages of manuscript in 100 working hours, and undertook to do so. After 36 working hours B was replaced by C, and the first half of the task was completed in 54 working hours from the start. After 80 working hours from the start B returned, and all three just finished the task in time. Find the average number of pages typed in an hour by A, by B, and by C.

GEOMETRY.

Tuesday, January 11- Afternoon, 2.50 to 5.50. 1. Prove that, if two sides of a triangle are unequal, the angle opposite the greater side is greater than the angle opposite the less. Also, state and prove the converse of this theorem.

In an acute-angled triangle ABC in which AB is greater than AC, the lines drawn from B, C perpendicular to the opposite sides intersect in 0. Prove that OB is greater than OC.

2. Prove that the diagonals of a rhombus bisect each other at right angles.

The side BC of a rhombus ABCD is produced through C to a point E so that CE is greater than BC. The line ED is drawn and produced to cut CA produced in F, and F is joined to B. Prove that the angles BFA, EFA are equal.

3. Prove that parallelograms on the same base and between the same parallels are equal in area.

ABCD, AEFG are two parallelograms having a common point at A, and having the vertex E on BC, and the vertex D on FG. Prove that the parallelograms are equal in area.

4. In a right-angled triangle prove that the square on the hypotenuse is equal to the sum of the squares on the other two sides.

If two unequal right-angled triangles ABC, ADC are drawn on opposite sides of their common hypotenuse AC, and if AM, CN are drawn perpendicular to BD, cutting it in M, N, prove that BM+BN?=DMP+DN?.

5. If a straight line AB is bisected at C and produced to any point D, prove that CD=AD. BD+AC?.

Show how to find the position of D by a geometrical construction so that the rectangle AD. BD shall equal the square on AB.

6. Show, with proof, how to construct a square equal in area to a given triangle, illustrating your method by a well-drawn figure.

7. Prove that the angle at the center of a circle standing on a given arc is double any angle at the circumference standing on the same arc.

Prove that, if a square be described externally on the hypotenuse of a right-angled triangle, and the right angle be joined to the center of the square, the joining line will bisect the right angle.

8. The angles A, B of a cyclic quadrilateral ABCD are 110°, 85°, respectively, and AB subtends an angle of 65° at the point of intersection of the diagonals. Find the angles which the sides subtend at the center of the circle.

9. Prove that in equal circles (or the same circle) equal angles at the centers stand on chords which are equal.

Given the base BC and the vertical angle A of a triangle ABC, the angle A being acute; prove that if B M is drawn perpendicular to AC, cutting it in M, and if CN is drawn perpendicular to AB, cutting it in N, the line MN is of constant length.

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