Sidebilder
PDF
ePub

4. The perpendiculars let fall from the extremities of the base of an isosceles triangle on the opposite sides will include an angle supplementary to the vertical angle of the triangle.

5. Shew that the angles of an equiangular triangle are equal to two-thirds of a right angle.

6. Find the magnitude of the angle of a regular octagon. (Th. 26.)

7. How many equiangular triangles can be placed so as to have one common angular point, and fill up the space round it?

8. Shew that three regular hexagons can be placed so as to have a common point, and fill up the space round that point.

9.

Shew that two regular octagons and one square have the same property.

Draw a pattern consisting of octagons and squares.

IO.

Shew that the angle of a regular pentagon is to the angle of a regular decagon as 3 to 4.

II. If a line is perpendicular to another it will be perpendicular to every line parallel to it.

12. If a polygon is equilateral, does it follow that it is equiangular, and conversely?

13. How many diagonals can be drawn in a pentagon? How many in a decagon? How many in a polygon of n sides.

14. Shew that a square, a hexagon and a dodecagon will fill up the space round a point; and make a pattern of these polygons.

15. Examine whether a square, a pentagon and an icosagon have the same property; and also whether a pattern can be constructed of pentagons and decagons.

16. The exterior angle of a regular polygon is one-third of a right angle: find the number of sides in the polygon.

17. Two lines intersecting in A are respectively perpendicular to two lines intersecting in B: prove that any angle at A is equal or supplementary to any angle at B.

18. Shew that a trapezium may be divided into a parallelogram and a triangle.

19. The diagonals of any parallelogram will bisect one another.

20. The diagonals of a rhombus will bisect one another at right angles.

21. If two straight lines be drawn bisecting one another, and their extremities be joined, the figure so formed will be a parallelogram.

22.

Given that a four-sided figure has its opposite sides equal, prove that it must be a parallelogram.

23. Prove that the diagonals of a rectangle are equal to one another.

24. Shew that if one element (a side) is given, a square is determined; if two elements (a side and angle), a rhombus is determined; also that if two elements (two sides) are given, a rectangle is determined: and find the number of elements required to determine a parallelogram, a trapezium, a quadrilateral, a pentagon, and a polygon of any number (2) of sides.

QUESTIONS ON SECTION III.

1.

zium.

Give the derivation of the words parallel, parallelogram, trape

2. What is indirectly ascertained in Theorem 21? Would it be possible to ascertain it directly?

3. Prove Th. 24 by drawing a straight line to intersect A, B and X, and using Theorems 21, 22.

4. Given two angles of a triangle to be respectively 72°. 15.47" and 83°. 51′. 16", find the third angle.

5. If one angle of a triangle is equal to the other two, prove that it must be a right angle.

6. Isosceles triangles having equal vertical angles must have equal base angles.

SECTION IV.

PROBLEMS.

IN the Science of Geometry there are not only theorems to be proved, but constructions to be effected, which are called problems. Geometers have always imposed certain limitations on themselves with respect to the instruments which might be used in these constructions. There is no reason why any convenient instrument used in the Art of Geometry, such as the square, parallel ruler, sector, protractor, should not be supposed to be used also in the Science; but the ruler and compasses suffice for nearly all the simpler constructions, and those which cannot be effected by their means are considered as not forming a part of Elementary Geometry. These instruments are therefore postulated or requested (vid. p. 4). There are some problems, that seem at first sight not very difficult, that cannot be solved by the use of these instruments. We can, for example, bisect an angle; but we cannot, in general, trisect it, that is, divide it into three equal parts, by any combination of ruler and compasses.

It may be observed that the ruler is simply a straight edge, not graduated, and the compasses are supposed to be transferable from one part of the figure to another, the distance between the points being unaltered.

The solution of a problem in Elementary Geometry as above defined consists

(1) in indicating how the ruler and compasses are to be used in effecting the construction required;

(2) in proving that the construction so given is correct; (3) in discussing the limitations, which sometimes exist, within which alone the solution is possible.

We shall give several examples of such problems, and then discuss the principles of the methods we have used.

Construction.

PROBLEM I.

To bisect a given angle.

Let ABC be the given angle.

Take any equal lengths BA, BC, along its arms, and join AC.

With centre A, and any radius greater than half AC, describe a circle, and with centre C, and the same radius, describe another circle intersecting the former circle on the side of AC remote from B in D.

Join AD, CD, and BD;

BD bisects the angle ABC.

Proof. In the triangles ABD, CBD,

because

and BD is common,

AB = BC,

(Constr.)

and the base AD = the base DC, (Constr.) therefore the angle ABD = the angle CBD, that is, BD bisects the angle ABC*.

* Eucl. 1. 9.

B

A

(Th. 15.)

« ForrigeFortsett »