Euclid's Elements of Geometry

triangle to the base is less than the square on a side of a triangle by a rectangle contained by the segments of the base.

SECOND CLASS PROVINCIAL CERTIFICATES, 1875.

TIME-TWO HOURS AND THREE-QUARTERS.

NOTE. Those students who take only Book I, will confine themselves to the first seven questions. Those who take Books I and II, will omit the questions marked with an asterisk (*), namely, (1) and (2). **1. If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.

*2. If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, namely, the sides opposite to equal angles, then shall the other sides be equal, each to each.

3. If a straight line falling on two other straight lines make the alternate angles equal to each other, these two straight lines shall be parallel.

4. If a straight line fall upon two parallel straight lines, it makes the two interior angles upon the same side together equal to two right angles.

5. Assuming Proposition XXXII, deduce the corollary: "all the exterior angles of any rectilineal figure, made by producing the sides successively in the same direction, are together equal to four right angles."

6. If a straight line,drawn parallel to the base of a triangle, bisect one of the sides, it shall bisect the other also.

7. Let ABC and ADC be two triangles on the same base AC and between the same parallels. AC and BD. Prove, that, if the

sides AB and BC be equal to one another, their sum is less than the sum of the sides AD and DC.

8. If a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts are together equal to the square on the whole line.

9. If a straight line be bisected and produced to any point, the rectangles contained by the whole line thus produced, and the part of it produced, together with, etc. (6, II).

10. Divide a straight line into two parts, such that the sum of their squares may be the least possible.

FIRST CLASS PROVINCIAL CERTIFICATES. 1871.

TIME. THREE HOURS.

1. To describe a square that shall be equal to a given rectilineal figure. 2. A segment of a circle being given, to describe the circle of which it is

the segment.

3. If the vertical angle of a triangle be divided into two equal angles 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.

4. In a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are

similar to the whole triangle and to one another.

If four straight lines be proportionals, the similar rectilineal figures similarly described upon them shall also be proportionals.

Draw a straight line so as to touch two given circles.

7. Let A B C be a triangle, and from B and C, the extremities of the base B C, let line B F and C E be drawn to F and E, the middle points of A C and A B respectively, then, if B F C E, A B and A C shall be equal to one another.

8. Describe an equilateral triangle equal to a given triangle.

FIRST CLASS PROVINCIAL CERTIFICATES, 1872.

TIME-TWO AND A HALF HOURS.

1. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles which this line makes with the line touching the circle shall be equal to the angles which are in the alternate segments of the circle.

2. To inscribe a circle in a given triangle.

Equal triangles which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional.

4. Similar triangles are to one another in the duplicate ratio of their homologous sides.

In any right angled triangle, any rectilineal figure described on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle. Two circles cut each other, and through the points of section are drawn two parallel lines, terminated by the circumferences. Prove that these lines are equal.

7. Let A C and B D, the diagonals of a quadrilateral figure A B C D, intersect in E. Then, if A B be parallel to C D, the circles described about the triangles A B E and C D E shall touch one another.

8. Divide a triangle into two equal parts by a straight line at right angles to one of the sides.

FIRST CLASS PROVINCIAL CERTIFICATES, 1873.

TIME-THREE HOURS.

1. The angle in a semicircle is a right angle.

A segment of a circle being given, describe the circle of which it is a segment.

3. Give Euclid's definition of proportion; and prove, by taking equimultiples according to the definition, that 2, 3, 9, 13, are not proportionals.

4. Similar triangles are to one another in the duplicate ratio of their homologous sides.

5. To find a mean proportional between two given straight lines. 6. Through C, the vertex of a triangle A C B, which has the sides A C

and C B equal to one another, a line C D equal to one another, a line CD is drawn parallel to A B; and straight lines, A D, D B, are drawn from A and B to any point D in CD. Prove that the angle A C D is greater than the angle A D B.

7. A B C D is a quadrilateral figure inscribed in a circle. From A and B, perpendiculars A E, BF are let fall on CD (produced if necessary); and from C and D, perpendiculars C G, DH, are let fall on B A (produced if necessary). Prove that the rectangles A E, B F and C G, D H, are equal to one another.

8. ABCD is a quadrilateral figure inscribed in a circle. The straight line D E drawn through D parallel to A B, cuts the side B C in E; and the straight line A E produced meets D C produced in F. Prove, that if the rectangle B A, AD be equal to the rectangle E C, CF, the triangle AD F shall be equal to the quadrilateral A B C D.

FIRST CLASS PROVINCIAL CERTIFICATES, 1874.

TIME. THREE HOURS.

1. In equal circles, eqnal straight lines cut off equal circumferences, the greater, equal to the greater, and the less to the less.

2. To describe a circle about a given equilateral and equiangular penta

gon.

3. To find a mean proportional between two given straight lines.

What is meant by duplicate ratio? Write down two whole numbers, which are in the duplicate ratio of to }.

What are similar rectilineal figures?

Similar triangles are to one another in the duplicate ratio of their homologous sides.

5. In any right angled triangle, any rectilineal figure described on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle.

6. To describe a triangle, of which the base, the vertical angle, and the sum of the two sides are given.

From A the vertex of a triangle ABC, in which each of the angles ABC and ACB is less than right angle, AD is let fall perpendicular on the base BC. Produce BC to E, making CE equal to AD; and let F be a point in AC, such that the triangle BFE is equal to the triangle ABC. Prove that F is one of the angular points of a square inscribed in the triangle ABC, with one of its sides on BC.

8. Let E be the point of intersection of the diagonals of a quadrilateral figure ABCD, of which any two opposite angles are together equal to two right angles. Produce BC to G, making CG equal to EA; and produce AD to F, making DF equal to BE. Prove that if EG and EF be joined, the triangles EDF and ECG are equal to one another.

FIRST CLASS PROVINCIAL CERTIFICATES, 1875

TIME-THREE HOURS.

1. If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, namely, the sides adjacent to the equal angles in each, then shall the other sides be equal each to each.

2. From a given circle to cut off a segment, which shall contain an angle equal to a given rectilineal angle.

If the angle of a triangle be divided into two equal angles 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 triangles have to one another.

4. The sides about the equal angles equi-angular triangles are proportionals; and those which are opposite to the equal angles are homologous sides.

5. If the similar rectilineal figures similarly described upon four straight lines be proportionals, those straight lines shall be proportionals.

6. Any rectangle is half the rectangle contained by the diameters of the squares on its adjacent sides.

Through a given point within a given circle, to draw a straight line such that one of the parts of it intercepted between that point and the circumference shall be double of the other.

8. If, from any point in a circular arc, perpendiculars be let fall on its bounding radii, the distance of their feet is invariable.

MATRICULATION, 1871.

1. State the points of agreement and disagreement of the circle, square and rhombus, with one another as appearing from their definitions.

2. Any two sides of a triangle are together greater than the third side. Show that the sum of the excesses of each pair of sides above the third side is equal to the sum of the three sides of the triangle. 3. If the square described upon one of the sides of a triangle be equal to the square described on the other two sides of it, the angle contained by these two sides is a right angle.

In an isosceles triangle if the square on the base be equal to three times the square on either side the vertical angle is two-thirds of two right angles.

4. If a straight line be divided into any two parts the square on the whole line is equal to the square on the two parts, together with twice the rectangle contained by the parts.

Is there any difference between the principle of this proposition and the statement (a + b)2 = a2 + 2ab + b2.

Of all the squares that can be inscribed within another the least is that formed by joining the bisections of the side.

5. If a straight line be divided into two equal and also into two unequal parts, the squares on the two unequal parts are together double of the square on half the line and of the square on the line between the points of section.

Does the statement respecting the equality of the square hold for any other division of the line.

C. Equal straight lines in a circle are equally distant from the centre; and conversely, those which are equally distant from the centre are equal to one another.

The lines joining the extemities of two equal straight lines in a circle towards the same parts are parallel to each other.;

7. What is meant by the Angle in a segment of a circle? Define simil ar segments of circles.

Upon the same straight line and upon the same side of it, there can. not be two similiar segments of circles not coinciding with one another. 8. In equal circles the angles which stand upon equal arcs, are equal to one another whether they be at the centres or circumferences. If two equal circles so intersect each other that the tangents at one of their points of intersection are inclined to each other at an angle of 60° shew that

Radius of circle: line joining their centres: 1:√3.

9. From a given circle to cut off a segment that shall contain an angle equal to a given rectilineal angle.

In a given circle inscribe a triangle which shall have a given vertical angle and whose area shall be equal to a given triangle; and shew with what limitation this can be done.

10. When is a circle said to be inscribed in a rectilineal figure. To inscribe a circle in a given triangle.

11. Inscribe an equilateral and equiangular pentagon in a given circle. Show how to divide a right angle into fifteen equal parts.

MATRICULATION, 1872.

HONORS.

1. From a given point to draw a straight line equal to a given straight line.

Explain what different constructions there are in this proposition. 2. If a side of a triangle be produced, the exterior angle is equal to the two interior and opposite angles; and the three interior angles of every triangle are together equal to two right angles. Find the number of degrees in one of the exterior angles of a regular heptagon.

3. Triangles upon the same or equal bases and between the same parallels are equal to one another.

By means of these propositions prove that a line drawn parallel to the base of a triangle and cutting off one-fourth from one of its sides, will also cut off a fourth part from the other side.

4. If a straight line be divided into two equal and also into two unequal parts, the squares on the two unequal parts are together double of the square on half the line, and of the square on.the line between the points of section.

If a chord be drawn parallel to the diameter of a circle and from any point in the diameter lines be drawn to its extremities, the sum of their squares will be equal to the sum of the squares of the segments of the diameter.

5. To divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts shall be equal to the square on the other part.

Solve the problem algebraically. Interpret and construct geometrically the second root so obtained.

Divide a given line so that one segment may be a geometric mean between the whole and the other.

6. In every triangle, the square on the side subtending either of the acute angles, is less than the squares on the sides containing

« Previous Continue »