## 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 0. The line OP 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 O. 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 O P and P E 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 .1 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 O, P, and E.

JANUARY 14, /9/0. HERBERT CHANDLER, CH/CAGO, /LL . PLATE V.

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

## Plate V

Penciling. In laying out the border lines and center lines follow the directions given for Plate IV. Draw the dot and dash lines in the same manner, as there are to be six problems on this plate.

Problem 7. To draw a perpendicular to a line from a point without the line.

Draw the straight line A C about 3 1/4 inches long, and assume the point P about 1 1/2 inches above the line. With P as a center and any convenient radius - about 2 inches - describe an arc cutting A C at E and F. The radius of this arc must always be such that it will cut A C in two points; the nearer the points E and F are to A and C, the greater will be the accuracy of the work.

Now with E and F as centers and any convenient radius - about 2 1/4 inches - draw the arcs intersecting below A C at T. A line through the points P and T will be perpendicular to A C. In case there is not room below A C to draw the arcs, they may be drawn intersecting above the line as shown at N. Whenever convenient draw the arcs below A C for greater accuracy.

Proof. Since P and T are both equally distant from E and F, the line P T is perpendicular to AC.