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PARALLELOGRAM OF FORCES
If two adjacent sides represent in direction and amount two given forces acting at the point of meeting, a diagonal from this point will represent in direction and amount the equivalent force, or resultant of the two given forces.
The opposite sides of a parallelogram, as A B C D, are equal, also the opposite angles are equal; and the diameter or diagonal C B divides it into two equal parts.
Since A B and C D are parallel, and C B meets them, the angle A B C is equal to the alternate angle BCD (1,29). And because A C is parallel to B D and BC meets them, therefore the angle A C B is equal to the alternate angle C BD (1,29).
Hence in the two triangles A CB, CBD, because the angles A B C, BC A in the one are equal to the two angles B C D C B D in the other, each to each; and one side B C which is adjacent to their equal angles, common to the two triangles ; therefore their other sides are equal, each to each, and the third angle of the one equal to the third angle of the other (1,26), namely, the side A B to the side CD, and AC to BD, and the angle BAC to the angle BDC, and because the angle A B C is equal to the angle BCD, and the angle C B D to the angle A CB, therefore the whole angle A B D is equal to the whole angle A CD; and the angle B A C has been shown to be equal to BDC; therefore the opposite sides and angles of a parallelogram are equal to one another. Also the diameter B C bisects it. For since AB is equal to C D, and B C common, the two sides, A B, BC, are equal to the two DC, C-B, each to each, and the angle A B C has been proved to be equal to the angle BCD; therefore the triangle A B C is equal to the triangle B C D (1,4); and the diameter B C divides the parallelogram ACĎ B into two equal parts.-Q. E. D.
In a right-angled triangle the square described on the hypotenusə is equal to the sum of the squares described on the other two sides.
Let A B C be a right-angled triangle, having the angle B A C a right angle; then shall the square described on the hypotenuse B C be equal to the sum of the squares described on BA, A C.
On B C describe the square B D E C; and on B A, AC describe the squares BAGF,ACKH. Through A draw, AL parallel to BD or C E; and join A D, F C. Then because each of the angles B A C, BA G is a right angle, CA and A G are in the same straight line. For the same reason, B A and A H are in the same straight line. Now the angle C B D is equal to the angle F B A,
for each of them is a right angle. Add to each of these angles the angle A B C, therefore the whole angle D B A is equal to the whole angle F B C; and because the two sides A B, BD, are equal to the two sides FB, BC, each to each, and the included angle A B D is equal to the included angle FB C, therefore the base A D is equal to the base F C, and the triangle FBC to the triangle A B D.
Now the parallelogram B L is double of the triangle A B D (I. 41) because they are upon the same base and between the same parallels B D and A L; also the square G B is double of the triangle F B C, because these also are on the same base F B, and between the same parallels F B, G C.
But the doubles of equals are equal to one another; therefore the parallelogram B L is equal to the square B G. Similarly, by joining A E, B K, it can be proved that the parallelogram C L is equal to the square HC. Therefore the whole square B D E C is equal to the two squares G B, HC.-Q. E. D.
To find the Centre of a given Circle Let A B C be the given circle; it is required to find its centre
Draw within it any straight line A B and bisect A B at D; from the point D draw DC at right angles to A B, produce CD to meet the circumference at E and bisect CE at F. The point F shall be the centre of the circle A B C. It is manifest that if, in a circle, a straight line bisect another at right angles, the centre of the circle is in the straight line which bisects the other.
The angle at the centre of a circle is double the angle at the circumference on the same base, that is, on the same arc.
Let A B C be a circle and BEC an angle at the centre, BA C an angle at the circımference which have the same arc B C for their base; the angle B E C shall be double of the angle B A C. Let the centre of the circle be within the triangle B A C; join A E and produce it to F; then because E A is equal to EB, the angle E B A is equal to the angle E A B; the angle EB A plus the angle E A B equals the supplement of the angle A E B, but the angle B E F is also the supplement of A E B, therefore the angle BEF is double of the angle E A B. It can be shown that the angle F EC is double of the angle E A C; therefore the whole angle B E C is double of the whole angle B A C.
The angles on the same segment of a circle are equal to one another.
Let A B C D be a circle, and BAD, BED angles in the same segment, B AED; the angles B AD, B E D shall be equal to one another. Take F the centre of the circle ABCD.
Let the segment B A E D be greater than a semicircle. Join B F, D F.
Then because the angle B F D is at the centre and the angle B A D is at the circumference, and that they have the same arc for their base, namely BCD; therefore the angle BFD is double the angle B A D (M, 20). For the same reason the angle B F D is double the angle B ED.
Therefore the angle B A D is equal to the angle B E D.
Wherefore the angles in the same segment of a circle are equal to one another, whether the segment be greater or less than a semicircle.
The construction of triangles will be explained in Trigonometry.
PLANE TRIGONOMETRY Plane Trigonometry is that branch of Mathematics which teaches us how to compute the sides and angles of plane triangles ; it is divided into right-angled and oblique-angled Trigonometry, according as it is applied to the mensuration of right-angled triangles or oblique-angled triangles. It is used by the navigator to solve all the problems in the sailings except great circle sailing, which is solved by right-angled and oblique-angled spherical trigonometry.
RIGHT-ANGLED TRIGONOMETRY 1. Every triangle consists of six parts ; namely, three sides and three angles.
2. The sum of three angles of every plane triangle is equal to two right angles, or 180°; hence, if one of the angles be known, the sum of the other two may be found by subtracting the given angle from 180° : also, if two of the angles be known, their sum subtracted from 180° will give the third angle. Again, in a right-angled triangle (since the right angle contains 90°), the sum of the two acute angles is equal to 90° : therefore, if one of the acute angles be given, the other will be found by subtracting the given angle from 90°
3. Any two sides of a triangle added together will be greater than the third side.
4. The greatest side of a triangle is opposite the greatest angle, and the least side opposite the least angle; also, in the same triangle, equal sides are opposite to equal angles.
5. In any right-angled triangle, the side which is opposite to the right angle is called the hypotenuse ; and of the other two sides, one is frequently termed the base, and the other, the perpendicular.
In every right-angled triangle, the square of the hypotenuse (or side opposite to the right angle) is equal to the sum of the squares of the sides which contain that angle.
Hence also in a right-angled triangle, the square of either of the two sides is equal to the difference of the squares of the hypotenuse and the other side.
6. Two right-angled triangles are equal to one another in all respects, when they have
1. The hypotenuse, and a side of the one equal
to the hypotenuse and a side of the other,
each to each ;
5. A side and the opposite acute angle equal; 7. Two triangles are said to be similar when all the angles of the one are respectively equal to all the angles of the other; as, for instance, the triangle A B C is similar to the triangle DE F, because the angles A, B, and C are respectively equal to the angles D, E, and F.
The sides of similar triangles, opposite to equal angles, are proportional; thus in the triangles A B C and DEF, as A B is to DE, so is AC to DF, and so is BC to EF. Or as 4: 8:: 5:10 :: 3:6. But this is better illustrated on the basis of the ratio of the sides, as will be seen presently.
8. The old method of computing the sides and angles of a triangle was on the basis of proportion, and the names of the sides (as lines) bore reference to the arc of a circle and the angle that it subtended or measured. It is not wholly inappropriate to briefly note this method in connection with the annexed figure, bearing in mind that these definitions appertain to a quadrant, and an arc less than a quadrant.
Two arcs, the sum of which is a quadrant, or quarter of a circle, are called complements of each other, thus F C is the complement of E C, and EC the complement of F C.
In the quadrant ECF, let E C be an arc of a circle, and E A C the corresponding angle at the centre of the circle; then FC is the complemental arc, and FAC the complemental angle.
SINE.—The sine of an arc is a straight line drawn from one extremity of the arc perpendicular to the radius which passes through the other extremity of the arc; here C B is the sine of the arc E C, or of the angle E A C, to radius A C.
TANGENT.—The tangent of an arc is a straight line which touches the measuring arc at its commencement, and terminates in the radius produced through and beyond the other extremity of the arc : here E T is the tangent of the arc E C, or of the angle E A C, to radius A E, which is equal to A C.
SECANT.—The secant of an arc is a straight line extending from the centre of the circle, through one end of the arc, until it meets the further extremity of the tangent drawn from the commencement of the arc; here AT is the secant of the arc E C, or of the angle E AC, to radius A E or A C.
Since E is taken to be the origin or commencement of the arc which measures the angle E A C, so F may be taken to be the commencement of the arc F C which subtends the angle F AC; to the latter arc or angle, C D is the sine, F G is the tangent, and A G is the secant.
The complement of an arc being what the arc wants of a quadrant or 90°; in the figure, the arc C F is the complement of E C; and C A F, the angle which C F subtends, is the complement of the angle E A C. Now the sine, tangent, and secant of the complement of an arc are called the COSINE, COTANGENT, and COSECANT of the original arc.
Thus, generally, with respect to the arc E C or the angle E A C
CB is called the sine. abbreviated sin.
ver. sin., or vers.