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JUNE 1887.

(The candidate may omit any two propositions.)

1. To inscribe a circle in a given triangle.

2. If a perpendicular is drawn from the vertex of the right angle to the hypothenuse of a right triangle: (1st.) The two triangles thus formed are similar to each other, and to the whole triangle. (2d.) The perpendicular is a mean proportional between the segments of the hypothenuse. (3d) Each side about the right angle is a mean proportional between the hypothenuse and the adjacent segment.

3. The area of a circle is equal to half the product of its circumference by its radius.

4. Given the radius of a circle as 8 inches; find the circumference of the circle; also the area of a sector of 40° of the circle.

5. If BC is the base of an isosceles triangle, ABC, and BD is drawn perpendicular to AC, the angle DBC is equal to one-half the angle A.

6. AB is any chord and AC is tangent to a circle at A, CDE a line cutting the circumference in D and E and parallel to AB; show that the triangle ACD is similar to the triangle EAB.

JUNE 1888.

Do as many numbers as possible, at least No. 6 and three others. In the problems all constructions must be made with ruler and dividers, and sufficient indications of these constructions must be left on paper. The necessary proofs must accompany the solutions of problems, unless otherwise specified.

1. Given two concentric circles. All chords of the outer circle touching the inner one are of equal length.

2. Given a point and two intersecting lines; draw a line through the point terminated by the given lines in such a way as to be bisected at the point.

3. (a) Through a given point P without a given circle (center O) draw a tangent to the circle. (b) If the distance OP is 15 decimeters and the radius of the circle 9 decimeters, what is the length of the tangent from P to the circle?

4. Given two circles intersecting at L and M; and P any point on line LM produced. Prove: (a) the tangent lines. from P to the two circles are equal; (b) any point Q from which the tangents to the two circles are equal lies on the line LM.

5. If the line AL which bisects the vertical angle A of a triangle ABC, also bisects the base BC at L, then it is also perpendicular to the base.

6. (a) Define the meaning of 7; give also an approximate numerical value. (b) Give formulae for circumference and area of a circle with radius r. (c) What is ratio of radii of two circles whose areas are in ratio 81: 100?

7. Construct an equilateral triangle equal in area to the

square descrribed on a given line t. (Proof may be omitted.)

8. Construct lines x, y, z, u, determined by these proportions, in which t denotes a given line; (1) x: y: t = 5 :13; (2) zu: t=√5: 1:13. (Prcof may be omitted.)

9. (a) Define the converse of a proposition. (b) What is the converse of the converse of a proposition? Explain. (c) Give an instance of a geometric proposition and its converse which are both true. (d) Give an instance in which the proposition is true but its converse is not true.

JUNE 1889.

1. Solve one of the two following problems with ruler and compass. No explanation is to be written but each step of the construction is to be made clear on the figure. (a) Construct a right-angled isosceles triangle and inscribe a circle in it.

(b) Divide a straight line AB into three equal parts. Erect a square on the middle part and construct a triangle o equal area.

2. From two points, A and B, on an arc of a circle, straight lines AC, AD, BC, BD, are drawn to the ends of the chord of the arc. Find two similar triangles in the figure, prove them similar, and write the proportions of their homologous sides.

3. Prove that in equal circles two incommensurable arcs have the same ratio as the angles which they subtend at the center.

4. Prove that two triangles having a common angle are to each other as the products of the sides including the common angle.

5. Solve or prove one of the following propositions : (a) To circumscribe about a circle, a regular polygon similar to a given inscribed polygon.

(b) The homologous sides of similar regular polygons have the same ratio as the radii of their circumscribed circles, and their perimeters have the same ratio as these radii.

JUNE 1890.

1. Prove that the bisectors of the angles of a triangle meet in a point.

2. On a given straight line AB construct a segment of a circle containing an angle equal to a given acute angle C.

Show clearly on the figure the method of construction. Ruler and compass must be used. The proof is not required.

3. Prove that in any triangle the square on the side opposite an acute angle is equivalent to the sum of the squares on the other two sides diminished by twice the product of one of those sides and the projection of the other upon that side.

4. Prove that regular polygons of the same number of sides are similar polygons.

5. If the radius of a circle is 5 inches, compute its circumference and its area; also the perimeter, the area, and the apothem of an inscribed square.

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