GEOMETRY. 



63 



obtuse ; and the angle DAB, which in IOM than a right angle, 



1 1 . A plntie figure, in geometry, in a portion of a plane Burfaoe, 

 inclosed by one or moro linos or boundaries Tho hum .f nil 

 the boundaries is called the perimeter of tho figure, and tlu< )...r- 

 Hurfaco contained witliin tin- i>n 



' in a plane figure contained or bounded by a 



line, called the circumference or periphery, which is such 



that all Htrui^ht linoa drawn from a certain point within tho 



figure to the circumference are equal to each other. This point 



Fig. 6. 



fig. 7. 



is called tho centre of the circle, and each of tho straight lines is 

 called a radius of tho circle. Tho straight line drawn through 

 tho centre and terminated at both ends in the circumference, is 

 called the diameter of the circle. 



It is plain, from tho definition, that all the radii must be 

 equal to each other, that all the diameters must be equal to 

 each other, and that tho diameter is always double the radius. 

 In speaking or writing, tho circle ia usually denoted by three 

 letters, placed at any distance from each other, around the 

 circumference ; thus, in Fig. 7, the circle is denoted by the 

 letters A c B, or A B B ; or by any three of the other letters on 

 the circumference. Tho point o is tho centre ; each of tho 

 straight lines o A, O B, o c, o E, is a radius, and the straight 

 line A B is a diameter. 



13. An arc of a circle is any part of its circumference ; the 

 chord of an arc is the straight lino which joins its extremities. 



Fig. 8. 



Fig. 10. 



Fig. 11. 



14. A segment of a circle is the surface inclosed by an aro 

 and its chord. 



15. A sector of a circle is the surface inclosed by an arc, and 

 the two radii drawn from its extremities. 



Thus, in Fig. 7, the portion of the circumference A M c, 

 whose extremities are A and c, is an arc ; and the remaining 

 portion ABC, having tho same extremities, is also an arc ; the 

 straight lino A c is the chord of either of these arcs. The sur- 

 face included between tho arc A M c and its chord A c, is the 

 segment A H c ; there is also the segment ABC. The surface 

 included between the radii o c, o B, and the arc c B, is called 

 tho sector COB; the remaining portion of the circle is also a 

 sector. 



16. A semicircle is tho segment whose chord is a diameter. 

 Thus, in Fig. 7, A c B or A E B is a semicircle. The term 

 semicircle, which literally means half a circle, is restricted in 



B D 



Fig. 12. 



Fig. 13. 



geometry to the segment thus described ; but there are many 

 other ways of obtaining half a circle. 



17. Plane rectilineal figures are described under various 

 heads; as trilateral or triangular ; quadrilateral or quadrangular; 

 and multilateral or polygonal. 



18. A triangle (Figs. 8, 9, 10, and 11) is a plane rectilineal 

 figure contained by three straight lines, which aro called its 

 sides. No figure can be formed of two straight lines ; hence, 

 an angle is not a figure, its legs being unlimited as to length. 

 Triangles are divided into various kinds, according to tho 

 relation of their sides or of their angles : as equilateral (Latin. 



<BTUIU, equal, and latu,, a ride); Isoaeelea (Oreek. i**, equal, 

 and (bbw, a leg); and soaleoe (Greek, afcotfiKW, unequal). 



, , .. 



\n equilateral (eqnsl-cided) triangle u that whiob hes 

 thrc* equal rides (fig. 8). 



20. An uotedet (equal-logged) triangle is that which hs only 



tn , <-<|ii;il Hi'lcM (Kitf. 'j). 



'21. A tcalme (unequal) triangle u that whioh hM all iU 

 idea unequal (Fig. 10). 



22. A right-angled triangle u that whioh hM one of it* saglss 

 a right angle (Fig. 11), in which the angle at A w the right 



Fig. 14. 



Fi. 15. 



16. 



angle. The ride opposite to the right angle is called the 

 hypotenuse (the subtense, or line stretched under the right 

 angle), and the other two rides are called the bate "d the per- 

 pendicular ; the two latter being interchangeable according to 

 the position of the triangle. 



23. An obtuse-angled triangle is that which has one of iU 

 angles an obtuse angle (Fig. 10). 



24. An acute-angled triangle is that which ban all its angles 

 acute ; Figs. 8 and 9 aro examples as to the fr"gl*^ but UMTB 

 is no restriction as to the rides. 



In any triangle, a straight line drawn from the vertex of one 

 of its angles perpendicular to the opposite ride, or to that rid 

 produced (that is, extended beyond either of its extremities in 

 a continued straight line), is called the perpendicular of tho 

 triangle ; as in Fig. 12, where the dotted line A D ia the perpen- 

 dicular of the triangle ABC; and in Fig. 13, where the dotted 

 line o H drawn from the point a to the dotted part of the bass 

 produced is the perpendicular of the triangle fro. 



25. A quadrilateral figure, or quadrangle, is a plane rectilineal 



. 17. 



Fig. 18. 



Fig. 19. 



figure contained by four straight lines, called ita rides. The 

 straight line which joins the vertices of any two of its oppomte 

 angles, is called ita diagonal. Quadrangles are divided into 

 various kinds, according to the relation of their rides and 

 angles ; as parallelograms, including the rectangle, the square, 

 the rhombus, and the rhomboid ; and trapeziums, including the 

 trapezoid. 



26. A parallelogram is a plane quadrilateral figure, whose 

 opposite sides are parallel ; thus, Fig. 14, A c B D, is a parallelo- 

 gram, and A B, c D, aro its diagonals. 



27. A rectangle is a parallelogram, whose angles aro right 

 angles (Fig. 15). 



28. A square ia a rectangle, whose rides aro all TTT 1 

 (Fig. 16). 



Fig. 20. 



Fi*. 21. 



29. A rhomboid is a parallelogram, whoee angles are oblique. 

 The opposite angles of a rhomboid aro equal to one another 

 (Fig. 14). 



30. A rhombus, or lozenge, is a rhomboid, whose rides are all 

 equal (Fig. 17). 



31. A trapezium is a plane quadrilateral figure, whoee oppo- 

 site sides are not parallel (Fig. 18). 



32. A trapnoid is a plane quadrilateral figure, which has two 

 of its rides parallel (Fig. 19). 



33. A multilateral figure, or polygon, is a plane rectQiiwel 

 ficrnre, of any number of Fides. The term ia generally applied 

 t<> any fijrure whose rides exceed four in number. Polygons are 



