Supplementary Problems

Problem E - A Circle Contains 360 Degrees.

Draw any circle and with a T-square draw a horizontal diameter passing through point O, the center. At point O draw a vertical diameter. These four angles will be right angles. How many degrees will each contain?

Problem F - Draw a Right Angle (With the T-Square and Triangle).

Bi-sect this angle by method in Problem 6. How many degrees in each of the small angles? This angle is called a half-pitch cut (Chapter II, Par. 24. Chapter V, Par. 75).

Problem G - Draw a Circle With Point O as the Center.

Around point O how many 30-degree angles can be constructed? Draw them. How many 45-degree angles? Draw another circle and divide it into 45-degree angles.

Problem H - Draw an Equilateral Triangle, One Side Being Given.

Draw the line AB the length of the given side. With the compasses set to the radius AB, using A as a center, draw an arc. With the same radius, using B as the center, draw an arc intersecting the first arc. Letter this point of intersection C. Join C with A and with B. This will be the desired equilateral triangle.

Problem No. 9 - To Construct a Triangle Having Given One Side and the Two Adjacent Angles.

Let line AB be the given side, angle Cde and Fhg the two adjacent angles. Draw line XY equal to AB. Using X for the vertex, draw the angle Kxl equal to the angle Cde. Using Y for the vertex, draw the angle Myn equal the angle Fhg (this should be done by the method of transferring angles already given). Produce the sides of the angles thus constructed until they meet at point Z. Zxy will be the required triangle.

Problem No. 10 - To Inscribe a Circle in a Given Triangle.

Let Abc be the given triangle. In order to inscribe the circle it will be necessary to find the center. To do this bi-sect any two angles (by the process already learned). Bi-sect the angle Abc with the line HB. Bi-sect the angle Acb with the line GC. These bi-sectors will intersect at the point K. Using K as a center and a radius equal to the perpendicular distance to any side of the triangle, draw the required inscribed circle.

Problem No. 11 - To Circumscribe a Circle About a Given Triangle, Or to Describe An Arc Or Circumference Through Three Given Points Not in the Same Straight Line.

Let Abc be the vertices of the triangle (three points not in a straight line). Bi-sect any two sides of the triangle (by the process already learned). Side AB will be bi-sected by the line FG; side BC by the line DE. Produce these perpendicular bi-sectors until they intersect at point K. Point K is the center of the required circle. Using K as a center, with a radius equal to the distance AK, draw the required circle through A, B and C.

Problem No. 12 - To Construct an Equilateral Triangle Having the Altitude Given.

Let line AB be the given altitude. Through the extremities of this line draw parallel lines, CD and GH perpendicular to line AB. With A as a center, and any convenient radius, draw the semi-circle CD. With C and D as centers, using the same radius, draw arcs, cutting the semi-circle at points E and F, respectively. Draw AE and AF; produce them to cut the line GH. Agh is the required equilateral triangle.

Problem 9

Problem 10

Problem 11

Plate III.