First Method. When The Point Is Near The Middle Of The Line

Draw the line A C about 3 1/2 inches long and assume the point P near the middle of the line. With P as a center and any convenient radius - about 1 1/4 inches - draw two arcs cutting the line A C at E and F. Now with E and F as centers and any convenient radius - about 2 1/2 inches - describe arcs intersecting at O. The line OF will be perpendicular to A C at P.

Proof

The points P and 0 are both equally distant from E and F. Hence a line drawn through them is perpendicular to E F at P.

Second Method. When The Point Is Near The End Of The Line

Draw the line A C about 3 1/2 inches long. Assume the given point P to be about 3/4 inch from the end A. With any point D as a center and a radius equal to D P, describe an arc cutting A C at E. Through E and D draw the diameter E 0. A line from 0 to P is perpendicular to A C at P.

Proof

The angle OPE is inscribed in a semicircle; hence it is a right angle, and the sides OF and PE are perpendicular to each other.

Lettering

After completing these figures draw pencil lines for the lettering. Place the words "Plate IV" and the date and the name in the border, as in preceding plates. To letter the words "Problem 1," "Problem 2," etc., draw three horizontal lines 1/4 inch, 3/8 inch, and 7/16 inch, respectively, above the horizontal center line and the lower border line to serve as a guide for the size of the letters.

Inking

In inking Plate IV, ink in the figures first. Make the line A C, Problem 1, a full line as it is the given line; make the arcs and the line D E dotted as they are construction lines. Similarly in Problem 2, make the sides of the angles full lines and the chord L M and the arcs dotted. Follow the same plan in inking the lines of Problems 3, 4, 5, and 6. In Problem 6, ink in only that part of the circumference which passes through the points 0, P, and E.

After inking the figures, ink in the heavy border line, and the lettering.