XII. SUPPLEMENTARY CONSTRUCTIONS BASED ON CIRCLES 78. Constructing the perpendicular bisector of a segment with compasses and ruler. The perpendicular bisector of a segment is the perpendicular erected to the segment at its mid-point. EXERCISE 101 1. a. Draw a segment AB. b. With radius more than AB, and center A, draw a small arc above the center of AB, and another below the center of AB. c. With the same radius and B as center, draw small arcs cutting the two drawn previously, at points C and D. d. Draw CD, intersecting AB at E. e. Compare segments AE and EB. (See Ex. 2, p. 115.) E f. What kind of angles are AEC and CEB? (Test them.) g. Below, print: CD is the perpendicular bisector of AB. 2. Draw an oblique segment. Construct its perpendicular bisector. 3. Draw a vertical segment. Construct its perpendicular bisector. 4. Draw a horizontal segment. Divide it into four equal parts by means of three perpendicular bisectors. 5. a. Draw a reasonably large triangle. Construct the perpendicular bisector of each of its sides. What happens? b. Compare the distances to the point of intersection from each of the three vertices. 6. Repeat Ex. 5, for a triangle of different shape. Do you obtain the same result? 7. Draw any segment AB. Construct the perpendicular bisector of it. On this perpendicular bisector, take any point 0. Draw OA and OB and compare them. Do the same for two more points on the perpendicular bisector. What seems to be true? 79. Constructing the bisector of an angle with compasses and ruler. EXERCISE 102 1. a. Draw an acute angle AOB. b. With any radius, and O as center, draw an arc cutting OA at C and OB at D. B e. Compare E c. With a radius more than one half CD, and C as center, draw an arc inside the angle, near where the bisector must pass. With the same radius, and D as center, draw a second arc cutting the first one at E. d. Draw OE. AOE and EOB. What name is given to the line that so divides an angle? f. Below, print: OE bisects ZAOB. 2. Draw an obtuse angle and bisect it. 3. Draw a second obtuse angle and divide it into four equal parts. 4. Draw a triangle of reasonably large size. Construct the bisectors of each of its angles. What happens? 5. Repeat Example 4 for a triangle of different shape. What seems to be true? 80. Constructing the perpendicular to a line from a point not on the line by means of compasses and ruler. EXERCISE 103 1. a. Draw any line AB, and place a point C, above AB. b. With C as center and a radius long enough to cut AB, draw an arc cutting AB at D and at E. c. With a radius more than one half DE, and D as center, B A D E F draw an arc below the approximate center of DE. With the same radius, and E as center, draw a second arc, cutting the first one at F. d. Draw CF, cutting AB at G. e. Determine what kind of angles are formed at point G. f. Below, print: CF is perpendicular to AB from C. 2. Repeat Example 1 for an oblique line. 3. Repeat Example 1 for a vertical line. 4. Draw a triangle of reasonably large size. From each vertex, construct the perpendicular to the opposite side, as in Examples 1 and 2. What happens? 5. Repeat Example 4 for a triangle of different shape. What name is given to the line drawn from a vertex perpendicular to the opposite side? 81. Constructing the perpendicular to a line at a point on the line, by means of compasses and ruler. EXERCISE 104 1. a. Draw any horizontal line AB, and place point C on it. b. With any radius and C as center, draw two arcs, cutting AB at D and at E AT C c. With a radius more than one half DE, and D as center, draw a small arc over the approximate e center of DE. With the same radius and E as center, draw a second arc, cutting the first one at F. d. Draw CF. What kind of angles are formed at C? What kind of lines are AB and CF? e. Below, print: CF is perpendicular to AB at C. 2. Repeat Example 1 for a vertical line. 3. Repeat Example 1 for an oblique line. 4. Draw a horizontal line. Place on it three points. At each, draw a perpendicular to the line. What kind of lines do you obtain? Regular Polygons 82. Dividing a circle into equal arcs. EXERCISE 105 1. a. Draw a circle with radius 14 inches. Mark its center O. From O, draw two radii OA and OB, forming an angle of 40°; then draw radius OC, so that / BOC shall also be 40°. b. With tracing paper compare arc AB and arc BC. and C 40% B A c. Below, write: ZAOB Z BOC are central angles of the circle. When central angles are equal, they cut out from the circle arcs. Arc AB and arc BC each contain 40 arc degrees. 2. a. Draw a circle with radius 1 inches. At its center, draw three central angles, each containing 60°. Compare their arcs. b. In a second circle of the same size, see how many 60° angles can be drawn "side by side" around the center. Into how many equal parts do they divide the circle? c. Draw the chord of each of the arcs in part b. Compare them. d. Below, write: If arcs of a circle are of these arcs also are the chords 3. a. If you wish to divide a circle into four equal arcs, how many equal central angles must you have around the center? How large will each one be? b. Draw a circle with radius 2 in. Using your protractor, make the equal angles which you need around the center. Test the arcs you obtain. If they are not equal, repeat the drawing, until they are approximately equal. c. Draw the chords of these arcs. What kind of figure seems to be formed? Test its angles and sides. d. Below, write: A square inscribed in a circle. |