B

D

с

Fig. 248

128. Construction of perpendicular lines. Let it be required to construct a line perpendicular to the seg

ment AB (Fig. 248) at the point C.

Using C as center and a convenient radius, draw arcs at D and E.

Using D and E as centers and a radius greater than CE, draw arcs intersecting at F.

Draw FC.
Then FC is the required perpendicular to AB.

Test the accuracy of your construction with the protractor.

We may prove that CF is perpendicular to DE by first showing ADCFYAECF.

It then follows that the two adjacent angles at C are equal.

.:CFIDE

EXERCISES

1. Construct angles of 90°, 45°, and 135°, using only compass and straightedge. 2. Construct a square, using only compasses and straight

edge.

B

3. Construct a square inscribed in a circle. Suggestion: Draw a diameter, as AB (Fig. 249).

Draw diameter CDLAB.
Draw AD, DB, BC, and CA.
ADBC is the required square.

Fig. 249

4. Construct Figs. 250 to 254, using only compass and straightedge. Make the diameter of the large circle 3 inches.

5. Draw a regular inscribed octagon (8-sided polygon).

Suggestion: Draw two diameters perpendicular to each other. Then draw the bisectors of the angles at the center intersecting the circle. Join these points of intersection.

Fig. 255

FIG. 256

6. Make the designs shown in Figs. 255 and 256.

129. Draw a perpendicular to a line from a point not on the line. Draw AB (Fig. 257). Select a point C not

on AB.

E

B

With C as center and a convenient radius, draw an arc intersecting AB at points D and E.

With D and E as centers, draw arcs intersecting at F.

Draw CF.

CF is the required line. FIG. 257

Test the accuracy of your construction with the protractor.

130. What is meant by distance from a point to a line. The length of the perpendicular from a point to a

line, as CG (Fig. 257), is called the distance from the point to the line.

C

From the point of intersection D of the bisectors of the angles draw DE perpendicular to AB.

Using DE as radius and D as center, draw a circle.

If the construction is carefully made, the circle just touches the sides of the triangle.

3. Draw a triangle ABC. Extend two of the sides, AB and AC, as shown in Fig. 260. Bisect the exterior angles at B and C and the interior angle at A.

Denote the point of intersection of these three lines by 0. Using O as center and the perpendicular from 0 to BC as radius, draw a circle. If the drawing is accurate, the circle

Fig. 260 touches BC and the extensions of AB and AC.

131. Altitude of a triangle. The perpendicular from the vertex of a triangle to the opposite side ($130) is an altitude of the triangle.

132. Concurrent lines. Lines passing through the same point are concurrent lines; e.g., the altitudes, the bisectors of the angles, and the perpendicular bisectors of the sides of a triangle are concurrent lines.

133. Constructing an angle equal to a given angle. Draw LABC (Fig. 261).

Draw a segment DE.

HL With B as center and any radius, draw

B4 arcs cutting BA and

F A BC at F and G respec

Fig. 261 tively.

With D as center and the same radius, draw arc HK.

iK

E

With K as center and radius equal to FG, draw an arc cutting HK at L.

Draw DL.
ZKDL is the required angle.

Measure both angles with the protractor to test the accuracy of the drawing.

To prove that ZB= _D, draw GF and LK.
Then prove AFGBY AKLD.

EXERCISES

1. Through B (Fig. 262) draw a line parallel to AC, using compass and straightedge only. See 870 for construction with

the protractor.

Suggestion: At B draw an angle equal to LA ($130).

2. Draw a triangle and bisect

one side. Through the mid-point, А

B
Fig. 262

draw a line parallel to another

side of the triangle. If the drawing is accurate, the third side will be bisected by the parallel. Use this as a test of accuracy.

THE CIRCUMFERENCE OF A CIRCLE

134. Measurement of the circle. With a tape line, measure carefully the distance around a cylindrical jar or tin can, and record the result in the second column of a table as shown below.

Measure the distance around other circular objects, such as plates or phonograph records, and place the results in the table.

Cut a circular piece of cardboard, and measure the distance around the circle. Enter the result in the table.