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The University of New Zealand.

Entrance Examination, 1885—Junior Scholarships.

Euclid and Trigonometry.

1. Prove that any two sides of a triangle are together greater than the third side.

If straight lines be drawn from the angular points of a triangle to the middle points of the opposite sides, those straight lines are together less than the perimeter of the triangle.

2. If the middle points of the opposite sides of any quadrilateral be joined, the squares on these lines are together half the squares on the diagonals.

3. Prove that in a circle equal chords cut off equal arcs.

The side BC of a triangle is bisected at right angles by a line which meets the circumscribing circle in D and E, prove that AD and AE are the bisectors of the internal and external angles at A.

4. Inscribe a circle in a given triangle.

5. Triangles and parallelograms of the same altitude are to one another as their bases.

ABC is a triangle, D the middle point of BC, any line through B meets AD in E and AC in F; prove that the ratio of CF to FA is twice that of DE to EA.

6. Prove that the formula cos (90° + A) = − sin A.

Show that

7. Prove that cos (⍺ + β) = cos ⍺ cos β−sin ⍺ sin β when ⍺ and β are each less than a right angle but their sum greater than a right angle.

Prove that

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8. Find a formula which will include all the angles which have the same tangent as a.

Solve the equation .

9. Prove that in any triangle

10. An artillery officer knows that the distance between two of the enemy's forts is c, and that the line joining them runs east and west. He finds that the forts subtend an angle a at his eye, and, after walking a distance ⍺ parallel to the line joining the forts, he finds that they again subtend an angle ⍺. Show how to find his original distance from the nearest fort.