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(13) Define similar triangles; can triangles be similar and not equal? Can they be equal and not similar? Explain fully.
(14) Are all equilateral triangles similar? Are two isosceles triangles necessarily similar?
(15) If each of the sides of a triangle be bisected, shew that the lines joining the points of bisection will divide the triangle into four equal triangles similar to the whole triangle and to each other.
(16) If through the vertex of each angle of a triangle a straight line be drawn parallel to the opposite side, shew that these lines will form a triangle similar to the given triangle; and find the ratio of this triangle to the given triangle.
(17) If the sides of any quadrilateral figure be bisected, shew that the lines joining the points of bisection will form a parallelogram.
(18) Shew that any triangle cut off from an equilateral triangle by a line parallel to one of its sides is equilateral.
(19) Through a given point draw a straight line, terminated by two given straight lines, so that it shall be bisected in that point.
(20) Through a given point draw a straight line, terminated by two other given straight lines, so that it shall be divided by that point in a given ratio.
(21) Of all triangles with two given sides shew that that is the greatest in which the two sides form a right angle.
(22) If an angle of a triangle be bisected by a straight line which also cuts the opposite side, shew that the two parts into which this side is divided will be in the same ratio as the other two sides are to one another.
(23) Shew that any two right-angled triangles are similar, if two of their acute angles, one in each triangle, are equal.
(24) If two triangles have the sides of the one, or sides produced, respectively at right angles to those of
the other, each to each, shew that the triangles are similar.
(25) If each of the sides of a triangle be bisected, and straight lines be drawn from the points of bisection to the vertex of the opposite angle, shew that these three lines will intersect in one point, and that the point of intersection divides each line into two parts of which one is double the other.
(26) In the last problem shew that the three lines from the point of intersection to the vertices of the three angles divide the given triangle into three equal triangles.
(27) Shew that two isosceles triangles will be similar, if any angle of the one be equal to the corresponding angle of the other.
(28) Find the greatest 'mean proportional' between any two lines of given sum.
(29) If two circles touch each other, either internally or externally, and two straight lines be drawn through the point of contact; so as to form four chords, two in each circle, shew that the four chords are proportionals.
(30) If two circles touch each other externally, and a straight line be drawn touching both and terminating at the points of contact, shew that this line is a mean proportional between the diameters.
(31) Shew that the parts into which the diagonals of a trapezium are divided by their point of intersection are proportionals.
(32) Shew that any rectangle is a mean proportional between the squares of two of its adjacent sides.
(33) Shew geometrically that a side of a square and its diagonal are incommensurable'.
(34) If on the sides of a right-angled triangle, taken as bases, three similar rectangles be described, shew that the rectangle on the side opposite to the right angle is equal to the sum of the other two.
POLYGONS, AND THEIR CONNECTION WITH THE
85. DEFINITION. A POLYGON* is a plane surface bounded by more than four straight lines, which are called its sides.
A plane surface with three sides has already received the name of triangle, and with four sides 'parallelogram', 'square', 'quadrilateral, or trapezium', as the case may be; therefore polygons begin with five sides, and may have any greater number.
An angle of a polygon means an angle formed by two adjacent sides of the polygon. And the number of the angles is obviously equal to the number of the sides.
DEF. A Polygon of 5 sides is called a Pentagon,
and so on.
DEF. A Regular Polygon is a polygon which has all its angles equal and all its sides equal.
Thus a regular Pentagon, Hexagon, and Octagon will respectively present the following appearance as to
[It does not yet appear that a regular polygon, as here defined, is a possible construction. All that is meant is, that, if such be possible, these are the distinctive names of such polygons §.]
DEF. The sum of all the sides of a polygon is called its perimeter.
*Polygon, derived from two Greek words, literally means a figure which has many corners.
Pentagon, that is, a five-cornered figure.
Hexagon, that is, a six-cornered figure.
Octagon, that is, an eight-cornered figure.
A similar observation might have been made, when the Definitions of equilateral triangle, and of a square, were given. We were not then able to say, that such constructions were possible.
DEF. A straight line. drawn from the vertex of any angle to the vertex of any other angle not adiacent to the former. is called a diagonal.
Thus, in the annexed ig ABBC - CD - DE - El is the vertmeter, and each of the straight lines AC, AD, BE, BD, ZC, is a tiagonal, of the poingon ABCDE.
N. B. Throughout this section all those polugoms are excluded which have what are cailed re-entrant angles, such as the polygons annexed:
where A, B, C are re-entrant angies. They are called re entrant angles, because if the lines forming them be produced through the vertex, these lines enter within the polygon, which is not the case with ordinary polygons.
88. Pane. I. All the angles of a polugon are toge ther equal to twice as many right angles as the polygon has sides, diminished by four right angles.
For every polygon, as ABCDEF, may be divided into triangles by taking any point O within the polygon, and joining 04, OB, OC, OD, OE, OF; and the number of triangles will ob- T viously be the same as the number of the sides of the polygon. But the three angles of each triangle are together equal to two right angles; the angles of all
the triangles are together equal to twice as many right angles as the polygon has sides; that is, all the angles of the polygon, together with the angles having the comThon vertex O, are equal to twice as many right angles the polygon has sides. But the angles at O are
equal to four right angles (30 Cor.); .. all the angles of the polygon are equal to twice as many right angles as the polygon has sides, diminished by four right angles. COR. 1. Hence, all the angles of a pentagon = 6 right angles; hexagon = 8 = 8 ....
and so on, whatever be the number of sides of the polygon*.
Hence, also, since all the angles are equal to one another in a regular polygon,
each angle of a regular pentagon =
of a right angle;
and so on.
COR. 2. If ABCDEF be a portion of the perimeter
of any polygon; and if the sides AB, BC, CD, &c., be produced to b, c, d, &c., since each interior angle, as ▲ ABC, + its exterior angle, as bBC, two right angles, .. all the interior angles + all the exterior angles = twice as many right angles as the polygon has sides; and .., by what has been proved, all the exterior angles of a polygon are together equal to four right angles.
The same result may also be made to appear from a very simple consideration. From B draw Bz parallel to CD, By parallel to DE, Br parallel to EF, &c., taking every side of the polygon in succession. Then DCc= LCB2, LEDd=<yBz, FEe = LxBy, &c.; and the last of the lines Bz, By, Bx, &c., will be Bb; .. the sum of
* The triangle, and quadrilateral, as we might expect, both follow the same rule. Thus all the angles of a triangle are equal to 6 right angles diminished by 4 right angles, that is, are equal to 2 right angles. And all the angles of a quadrilateral are equal to 8 right angles dimi. nished by 4 right angles, that is, are equal to 4 right angles.