Right triangle

“ For example, if two right triangles have equal sides, they are congruent; if two right triangles have two corresponding sides equal, they are congruent. Triangles are congruent only if the sides and angles are equal.
The sum of two sides of a plane triangle is always greater than the third side; their difference is always less than the third side. ”

“ A formula such as the equality of the interior angles of a triangle to two right angles is only scientifically known when it is not of isosceles or scalene triangle that it is known, nor even of all the several types of triangle collectively, but as a predicate of triangle recognized as the widest class-concept of which it is true, the first stage in the progressive differentiation of figure at which it can be asserted. ”

“ This process can be shortened by omitting the squaring of the hypotenuse, and just dividing the square of the base by the hypotenuse, with the remainder then subtracted by the hypotenuse, as is known in algebraic math. Half of this remainder is then the length of the vertical, as is shown in this formula:
( (a^2) /a) - (b^2) = ( (b^2) /a) -a
Section 120 All this applies to a right triangle. If the triangle is not a right triangle but an equilateral triangle there is only a need to know the measure of one side, and there is no need to know any of the angles. ”

“ The representation of a triangle is always limited to a certain size and figure. When we imagine a triangle, it is always with such or such extension and with greater or smaller angles. The imagination representing an obtuse angled triangle sees something very different from an acute or right angled triangle. But the idea of the triangle in itself is not subject to any particular size or figure ”

“ The conception of the triangle, with all the mathematical truth which it contains, is necessary, universal, and eternal. It has, therefore, its foundation in necessary being, or in the divine intelligence. But it is in God in an eminent mode, and formally only in the human intellect. Geometrical truth is founded in the essence of God, who is the archetype of the triangle and of every other geometrical figure. But that which the triangle imitates the human intellect cannot see ”

“ Thus, should we mention the word triangle, and form the idea of a particular equilateral one to correspond to it, and should we afterwards assert, that the three angles of a triangle are equal to each other, the other individuals of a scalenum and isosceles, which we overlooked at first, immediately crowd in upon us, and make us perceive the falsehood of this proposition, though it be true with relation to that idea which we had formed. If the mind suggests not always these ideas upon occasion, it proceeds from some imperfection in its faculties ”

“ He is a scalene triangle, and must be judged, accordingly, upon irregular, lop-sided principles; whereas you and I, commonplace mortals, are produced, like the earth, which is our preponderating element, with our triangles all right-angled, comfortable, and complete—for which blessing let us thank Providence, and be charitable to those who are necessarily windy and gaseous, from that unlucky scalene triangle upon which they have had the misfortune to be constructed, and which, you perceive, is quite at variance with the mathematical constitution of the earth! ”

“ With us the representation of the triangle is almost always simultaneous, excepting the case of exceedingly large triangles, much larger than we usually see, in which case, especially when we are unaccustomed to consider such, it seems necessary to go on extending the lines successively. ”

“ What more easy than for any one to look a little into his own thoughts, and there try whether he has, or can attain to have, an idea that shall correspond with the description that is here given of the general idea of a triangle—which is neither oblique nor rectangle, equilateral, equicrural nor scalenon, but all and none of these at once? ”

“ The young reader who knows nothing of the elements of geometry will get some idea of the fascinating character of that science. The triangle ABC in Fig. 27 is what we call a right-angled triangle, because the side BC is at right angles to the side AB. Now if we build up a square on each side of the triPg 32angle, the squares on AB and BC will together be exactly equal to the square on the long side AC, which we call the hypotenuse. This is proved in the case I have given by subdividing the three squares into cells of equal dimensions. ”

“ It will be observed that there are three figures corresponding to three cases which may exist; for the straight line drawn parallel to one side may cut the other sides, or may cut the other sides when they are produced through the extremities of the base, or may cut the other sides when they are produced through the vertex. In all these cases the triangles which are shewn to be equal have their vertices at the extremities of the base of the given triangle, and have for their common base the straight line which is, either by hypothesis or by demonstration, parallel to the base of the triangle. ”

“ Of course, all the men must be used on every occasion.
137.—A STUDY IN THRIFT.
Certain numbers are called triangular, because if they are taken to represent counters or coins they may be laid out on the table so as to form triangles. The number 1 is always regarded as triangular, just as 1 is a square and a cube number. Place one counter on the table—that is, the first triangular number. Now place two more counters beneath it, and you have a triangle of three counters; therefore 3 is triangular. Next place a row of three more counters, and you have a triangle of six counters ”

“ II It is a common custom to refer to the usual complication between one man and two ladies, or one lady and two men, or a lady and a man and a nobleman, or—well, any of those problems—as the triangle. But they are never unqualified triangles. They are always isosceles—never equilateral. So, upon the coming of Nevada Warren, she and Gilbert and Barbara Ross lined up into such a figurative triangle; and of that triangle Barbara formed the hypotenuse. ”

“ It will now be understood when I say that one of the keys to the puzzle was the fact that these are always the squares of triangular numbers—that is, the squares of 1, 3, 6, 10, 15, 21, 28, etc., any of which numbers we have seen will form a triangle.
Every whole number is either triangular, or the sum of two triangular numbers or the sum of three triangular numbers. That is, if we take any number we choose we can always form one, two, or three triangles with them. ”

“ Two triangles are all that are absolutely necessary for a beginner. The first is that shown in Figure 3, [20] which is called a triangle of 45 degrees, because its edge A is at that angle to edges B and C. That in Figure 4 is called a triangle of 60 degrees, its edge A being at 60 degrees to B, and at 30 degrees to C. The edges P and C are at a right angle or an angle of 90 degrees in both figures; hence they are in this respect alike. By means of these triangles alone, a great many straight line drawings may be made with ease without the use of a drawing square ”

“ Rectilinear figures cannot be divided into triangles and quadrilaterals because there are rectilinear figures which have more than four sides. On the other hand, triangles can be divided into equilateral, isosceles and scalene, since no other kind of triangle can exist. ”

“ It may now be pointed out that the triangle may be used to divide circles much more quickly than they could be divided by stepping around them with compasses. Suppose, for example, that we require to divide a circle into eight equal parts, and we may do so as in Figure 172, line a being marked from the square, and lines b, c and d from the triangle of forty-five degrees; the lines to be inked in to form an octagon need not be pencilled, as their location is clearly defined, being lines joining the ends of the lines crossing the circle, as for example, lines e, f. ”

“ No image could ever be adequate to our conception of a triangle in general. For the generalness of the conception it never could attain to, as this includes under itself all triangles, whether right-angled, acute-angled, etc., whilst the image would always be limited to a single part of this sphere. The schema of the triangle can exist nowhere else than in thought, and it indicates a rule of the synthesis of the imagination in regard to pure figures in space. ”

“ If we extend these lines until they intersect, we will have the required triangle, and can measure the two sides, which will be found to be about 12 inches and 8 inches long, and the third angle will measure just 26 degrees. It doesn’t make any difference on what scale we draw the triangle, whether it be miles, yards, feet, inches or fractions of an inch, the proportions will be the same. If the base line had been 6 half-inches, or 3 inches long, and the same angles were used, the other two lines would measure 12 half-inches, or six inches, and 8 half-inches, or 4 inches. ”

“ If a minor triangle is made which has an obtuse angle and three unequal sides, then again the sides of the large triangle cannot be equal. For example, if the first side of the minor triangle is six feet long, the second three feet, and the third four feet, and the cord along the side of the greater triangle one hundred and one times six feet, that is, one hundred and one fathoms, the distance between the mouth of the tunnel and the bottom of the last shaft will be a length one hundred times three feet, or fifty fathoms ”

“ The beginner, in order to avoid error, often finds it useful to draw several triangles, as unlike each other as possible, in order to make sure that his reasoning is equally applicable to all of them. But a difficulty emerges as soon as we ask ourselves how we know that a thing is white or a triangle. If we wish to avoid the universals whiteness and triangularity, we shall choose some particular patch of white or some particular triangle, and say that anything is white or a triangle if it has the right sort of resemblance to our chosen particular. ”

“ Draw an isosceles triangle whose equal sides shall each be less than the third side.
Write its two names within it.
Draw an oblique straight line twice as long as any short measure or unit.
At one end draw a straight line perpendicular to it, and three times as long as the same measure. ”

“ The triangulum characteristicum is similar to the triangle formed by the tangent, the ordinate of the point of contact, and the sub-tangent, as well as to that between the ordinate, normal, and sub-normal. ”