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UNIVERSITY OF LONDON
MATRICULATION EXAMINATION
SEPTEMBER, 1950
ELEMENTARY MATHEMATICS II
GEOMETRY
Ordinary Paper
Examiners:
H. Russell Davis, Esq., M.Sc.
A. Geary, Esq., M.A., M.Sc.
WEDNESDAY, September 13 Morning, 10 to 1
[Candidates must not attempt more than SEVEN questions.]
1. (i) A triangle ABC has AB = AC and (A = 34(. The external bisector of (B meets AC produced at D. Calculate (BDA.
(ii) The diagonals AC, BD of a cyclic quadrilateral ABCD meet at E. If (ABC = 125(, (ADB=37( and (CED = 68(, calculate (BDC and (ABD.
2. (i) A point T lies on the tangent at A to a circle of which AC is a diameter; TC cuts the circle atB. If AC = 8 in., AT = 6 in., calculate the length of BC and the area of triangle ABC.
(ii) A line parallel to the base BC of a triangle ABC cuts AB, AC at P and Q respectively. If AB=15 in., AC = 20 in., PB = 6 in., calculate the length of AQ and the ratio of the areas of the triangles APQ, ABC.
3. Construct a quadrilateral PQRS in which PQ = 3 in., PR = 4 in., RS = 2 EMBED Equation.3 in., PS = 2 in. and (Q= 75(.
Construct geometrically a triangle equal in area to the quadrilateral and calculate the area of the triangle by making suitable measurements in the figure.
4. Assuming that the opposite sides of a parallelogram are equal, prove that the diagonals bisect each other.
The diagonals of a parallelogram ABCD intersect at O. Points H, K are taken in AB, BC respectively; HO, KO are produced to meet CD, DA at L, M respectively. Prove that HKLM is a parallelogram.
5. Prove that the angle which an arc of a circle subtends at the centre is double that which it subtends at any point on the remaining part of the circumference.
A radius OC of a circle, centre O, is at right angles to a diameter AD. The line joining D to B, any point on the minor arc AC, cuts OC at E. Prove that ABEO is a cyclic quadrilateral and that AC bisects (BAE.
6. Construct a triangle ABC in which AB = 2 in., AC = 1 EMBED Equation.3 in., BC = 1 in. With centre A draw a circle of radius 1 in. to cut AC at D.
Construct geometrically
(a) the locus of points equidistant from B and D;
(b) a circle to touch the circle, centre A, at D, and to pass through B.
Measure and write down the radius of this circle.
7. Prove that the tangents drawn to a circle from an external point are equal.
The tangents at the points B and C on a circle meet at A. The bisector of (BAC meets the minor arc BC at P. Prove that BP and PC are equal and that P is the centre of the inscribed circle of triangle ABC.
8. Prove that the areas of similar triangles are proportional to the squares on corresponding sides.
A point C is taken on a circle of which AB is a diameter and BC is produced to meet the tangent at A to the circle at T. Prove that EMBED Equation.3 .
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