68. As the altitudes are not given, the areas of the figures drawn by different pupils should vary. The protractor or the triangle should be used in drawing the altitudes, which should then be carefully measured.

69. See 62, making the upper side 21 in. The fourth trapezoid in Arithmetic, Art. 929, 6, shows the manner of determining the dimensions of the required rectangle.

1267. 78. The diameters of these circles should be such as not to admit of the use of the protractor in drawing them. To ascertain the extremities of an arc of 120°, two lines are drawn meeting in the center of the circle at an angle of 120°. The portion of the circumference intercepted by these lines will constitute an arc of 120°. The remainder of this circumference will form an arc of 240°.

79. The pupil should be permitted to make the first attempts in his own way. He will doubtless soon discover that the distance between the points of his compasses in drawing the circle, is the length of the chord required, and that by placing one point of the compasses on any portion of the circumference of the circle just drawn, the other point will indicate on the circumference the other extremity of the chord.

To measure the length of the arc in degrees, draw radii to its extremities, and use the protractor to determine the angle made by these radii, which may be produced, if necessary. If the work is properly done, the angle should measure 60°,

which is the length of the arc.

80. Draw two light lines meeting at an angle of 72°. Using the vertex of the angle as a center, and any radius, draw an arc between the lines. This arc will measure 72°. Darken the lines from the arc to the vertex

to show the radii of the circle of which the arc is a part.

81, 82. Figs. 8 and 9 show sectors of 90° and 270°, respectively; Figs. 10 and 11, segments of 120° and 240°, respectively.

1268. 86. A diameter will divide the circle into two equal parts. A second diameter perpendicular to the first, drawn by means of the protractor or the triangle, will divide the circle into four equal parts.

To divide the circumference into four equal parts, it will not be necessary to draw the diameters. When he has the ruler in the proper place to draw the first diameter, the pupil needs to mark only the two points where the ruler cuts the circumference. The third point can be indicated when the triangle is placed at the center of the circle and against the ruler; etc.

While the scholars may be permitted in the beginning of this work to draw a number of unnecessary lines, and while it may be an advantage to even require it, they should gradually learn to make as few lines as possible. The construction lines that are employed, should be drawn very lightly and should not be erased. Other lines should be made more conspicuous. Careful pupils may be allowed to use ink for this purpose.

1269. It is not intended that the methods here suggested should be communicated in advance to pupils. Each should be allowed to try the problem in his own way. The discussion of the method employed afterwards on the blackboard, will suggest other and possibly better modes of procedure.

88. To inscribe a regular pentagon in a circle, it will be necessary to divide the circumference into five arcs of 72° each. The

protractor should be used to obtain the first arc; the remaining ones can be set off by the compasses, the first being used as a measure. Careful work should be exacted by the teacher.

89. Many pupils will have learned in their drawing lessons. the regular method of inscribing a hexagon in a circle. Those unfamiliar with this way, should not be shown it until after they have constructed the hexagon by means of the protractor.

It is a pedagogical mistake to suggest "short-cuts" to pupils before they thoroughly understand a general method. For this reason, the teacher should permit the members of her class to use the protractor in the construction of the inscribed triangle, leaving it to themselves to discover a simpler way. She should encourage, also, the employment of a variety of methods even if some of them are not very direct. The experiments made by pupils to discover a new mode of constructing a polygon, will help them in their geometrical study.

The chord of 60° being equal in length to the radius (79), the shortest method of inscribing a hexagon is to apply the radius as a chord six times. Two of these divisions of the circumference will make arcs of 120°, the chords of three of these forming the sides of an inscribed equilateral triangle.

90. Each of the six angles at the center contains 60°. Since the two sides AC and AB, enclosing any central angle, are radii of the circle, and therefore equal to each other, the angles oppo

site those sides are equal; that is, angle Aangle B. The ⚫angle at C being 60°, A+B= 180° — 60° = 120°, and A=B= 60°. Angles 1 and 2 (see Arithmetic), therefore, measure 60°

each, and the whole angle contained between two adjoining sides of the hexagon, measures 120°. After determining by this method the number of degrees in each angle of a regular hexagon, the pupils should be required to construct one, and to mark in each angle its contents in degrees, as in Fig. 13, verifying the result by using the protractor.

91. A careless scholar, measuring the number of degrees in each angle of a regular pentagon (Arithmetic, Art. 1268), will sometimes read from the wrong row of numbers on the protractor, getting the result 72°, instead of the correct one of 108°. As there are 72° in each division of the circumscribing circle, he will have no doubt of the correctness of his answer, unless he has been trained to estimate the size of an angle. In this case, he will see that each angle of a regular pentagon is obtuse, and, therefore, greater than 90°.

One method of calculating the number of degrees, is to divide the pentagon into five equal triangles, one of which is shown in Fig.

A

54°

C

B

FIG. 14.

14. The angle at Cis 72°. The sides CA and CB, being radii, are equal; which makes equal angles at A and B, each of which is of (180° — 72°), or 54°. Each of these angles is the half of one of the angles of the pentagon, so that these latter angles measure 108° each.

92. A circumscribed square touches the circle at four points, each side constituting a tangent. A tangent being perpendicular to the radius drawn to the point of contact, the square may be constructed by drawing perpendiculars to two diameters intersecting at right angles, using the triangle or the protractor for the purpose. The ingenious pupil will discover other ways; drawing, for instance, at each extremity of the two diameters, a line parallel to the intersecting diameter, by means of the ruler and the triangle; etc. No method should be permitted that merely approximates accuracy, such as determining that a line is parallel or perpendicular by the eye alone. The

average class will contain many members intelligent enough to pass upon the correctness of a given method, and they should be called upon to give reasons for any criticisms they may have to offer.

M

C

N

In circumscribing some polygons, a hexagon for instance, many pupils prefer locating points X and Y, instead of using the triangle and the ruler to draw a tangent XF. After dividing the circumference (Fig. 15) into six equal parts at 1, 2, 3, 4, 5, and 6, they draw a diameter from 1 to 4. Through 2 and 6 they draw a secant XA, making MX and MA each equal to the radius. 5 a second secant is drawn, and NY and NB are also made equal to the radius. A line drawn through X and Y will form one side of the circumscribed hexagon. One extremity of this side can be determined by producing the diameter 3C6 to F, and the other by a line through 2C5.

FIG. 15.

Through 3 and

NOTE. A secant is a line that cuts a circle at two points; a tangent is a line that touches a circle at one point.

The

93. The smallest number of triangles into which a pentagon can be divided, is three (Fig. 16). three angles of each triangle contain 180°, making the sum of angles 1-9, 540°. Since 1 A, 2 + 4 = B, 6 + 7 = C, 8

=

==

D; 3+5 E, the sum of the five equal

angles (A, B, C, D, and E) of a regular pentagon = 540°, and each equals 108°. This is the result that was found in 91 by another method.

94. The hexagon is divisible, as above, into four triangles, containing 180° × 4, or 720°; making each angle 720°÷6, or 120°.

95. A quadrilateral is divisible into 2 triangles; a pentagon,