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INDETERMINATE CASES.

423

Draw A D as before, make the angle at A equal to the given angle, and a c equal to 4 the greater of the given sides; and c is one point of the triangle. From c let fall the dotted perpendicular c e; and if c, the less of the given sides, is, less than c E, there can be no triangle, Let it be greater than ce, but less than c a; then, from c as a centre, with a radius equal to c, describe an arc; and this arc must cut A D, and will cut it in two points, B and B, on opposite sides of the point E, and equally distant from it. So that the triangle is either A B C, of which the side situated on a d is the smaller portion A B; or it is a B' c, of which the side situated in the line a d is the greater portion, A B', made up of the three parts, A B, B E, and E B'; and the difference of those two triangles is the triangle C BB', of which the sides c B and C B' are equal to each other, being radii of the same circle.

It is also evident that the arc described on c as a centre must cut the line A d between A and E; for if an arc is drawn about C, with the radius c A; the point B must be within the circle of which that arc is a part, because c b is less than c A the radius,

These four articles embrace all the data, furnished simply by the parts of a triangle, and without reference to any other figure, which apply generally to the construction of all triangles, whatever may be their form; and by which the perfect equality of any two triangles which possess

them

may be established. We have seen that all of them require some conditions or limitations, which are necessary before there can be a triangle : namely, that no one side shall be greater than the other two, and no two angles equal to or greater than two right angles. We have farther seen, that in the last of the four, it is required that the given angle shall be opposite to the greater of the two given sides; otherwise the triangle is either ambiguous, having two forms and values, or the data are in

424

EQUILATERAL TRIANGLE.

correct, and it is impossible. There is, indeed, one single case of the less side opposite the given angle in which the triangle is possible and not ambiguous; and that is, when the side opposite the given angle is exactly equal to the perpendicular c E, and the arc touches the line a D, but does not cut it. In this case, the angle opposite the greater given side is a right angle ; and therefore the three angles of the triangle are virtually, if not expressly, given; and therefore this is a peculiar case, and properly belongs to that class of data in which the three angles are given.

There are some particular triangles, however, which, from their simplicity or regularity, can be constructed with fewer data; and it may not be amiss to mention at least one or two of them. The first and simplest is an equilateral triangle, the only datum necessary for the construction of which is the length of one of the sides. For, if one side is given, all the sides are given; and all the angles are also given, for they are necessarily equal to each other, in consequence of the equality of the sides ; and each of them is the third part of two right angles, or the sixth part of four. There are some not unimportant conclusions to be drawn from this, to which we shall very briefly advert, after constructing the triangle.

Let A B be any straight line, it is required to construct an equilateral triangle, having each of its sides equal to A B.

B

Take another line equal to A B, and A and B as centres, with a radius equal to A B, describe arcs cutting each other in

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the point c; and join a c and B c; and a b c is the equilateral triangle required.

If each of the three angular points is made the centre, and an arc described upon each as a centre, meeting the other two, each of those arcs is the measure of the third part of two right angles; that is, of 60° of an equal circle, of which the side of the triangle is the radius; but as viewed from each angle, the side opposite that angle is also the chord of 600. Therefore, the chord of 600 is equal to the radius of a circle; and if the radius of a circle is applied six times to the circumference, it will subtend the whole circumference in six equal parts of 60° degrees each ; and if we join the adjacent points of division all round, we shall have inscribed a regular hexagon, or figure of six equal sides, within the circle.

The circumference of a circle can always be divided into two equal parts by drawing a diameter till it meets that circumference both ways; and the radius, applied as above stated, divides the circumference into six equal parts. Thus we have an easy means of dividing a circle into any number of parts, which number is either a multiple of six by two, or a quotient arising from the division of six by two. These matters are so simple, however, that any one may readily understand how they are to be done, without further explanation than has been now given.

This property of the chord of 600 being equal to the radius, affords us a very convenient method of erecting a perpendicular at any point of a given line, whether that point is or is not the extremity of the line. Thus, let A B be any line, and let it be required to erect at the extremity a of the line, another line perpendicular to A B.

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On a as a centre, with any radius not greater than a B-but the greater that it is the operation will be the more accurate, describe an arc, equal to at least a third of a circumference, from A B toward that side where the perpendicular is required. Apply this radius on the arc from the line A B to the point 1, and again from the point 1 to the point 2. Then from 1 and 2 as centres, and still with the same radius describe arcs cutting each other in the point c; join a c, and a c is perpendicular to a B at the point A. For the arc from the line A B to 1 is 60°, and so is the arc from 1 to 2. But the line c B bisects the arc 1, 2; and therefore the portion of the arc intercepted between A B and A c is 600 and the half of 60°, or 90°, which is the measure of a right angle ; therefore B A c is a right angle, and a c is perpendicular to A B.

This is one of the most convenient methods of obtaining a right angle in practice.

In the case of an isosceles triangle, or triangle having two equal sides, the only data required for the construction or for establishing the equality of two triangles, in the case of the data being the same, are the unequal side and one of the equal ones ; for if one of the equal sides is given, the other is given; and thus the problem resolves itself into the constructing of a triangle of which the three sides are given, and thus we have only to apply the equal side as a radius at each end of the unequal one, and describe arcs intersecting each other, in that direction in which we wish the triangle to be situated ; and when both

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case,

extremities are joined to the intersection of the arcs, the triangle is constructed. The only condition or limitation required in this

is that the equal side shall be greater than half the unequal one; because if it were not, the arcs would not intersect each other.

The next consideration with regard to triangles, is the equality of their surfaces when the sides and angles are different; but before we proceed to this, it will be desirable to consider some of the more elementary properties of

PARALLELOGRAMS, RECTANGLES, AND SQUARES, and the relations which they have to triangles.

A PARALLELOGRAM is a four-sided figure, of which the oppo, site sides are parallel; and a line joining the opposite angles either way, is called the diameter of the parallelogram. This figure is usually named by repeating four letters placed at the angles, and taken in regular order round it; or more briefly by two letters situated at opposite angles. In a parallelogram, every side is adjacent to two other sides, and opposite to the remaining one, that is, the one to which it is parallel ; and every angle is. adjacent to two angles, and opposite to the remaining one; also, there are two sides opposite to every angle. Therefore, though the sides of a parallelogram are given, the figure is not giventhat is, there is not data enough for the construction of it, unless at least one of the angles, or one of the diameters, is given. It will, therefore, be necessary to attend to some of the properties of the figure, before we proceed to the comparison of one parallelogram with another, or of parallelograms with tri, angles.

1. The opposite sides and angles of any parallelogram are equal to each other, and the diameter joining the opposite angles either

way

divides it into two equal parts,

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