CONSTRUCTION OF QUADRILATERALS

Construct a square, having given:

3. The sum of the diagonal and one side.

Construct a rectangle, having given:

4. Two adjacent sides.

5. One side and a diagonal.

6. One side and the angle between the diagonals. 7. The diagonals and the angle between them.

Construct a rhombus, having given:

8. The diagonals.

10. One diagonal and one angle.

9. One side and one angle.

Construct a parallelogram, having given:

11. One side and the diagonals.

12. The diagonals and the angle between them.

13. Two adjacent sides and one diagonal.

14. Two sides and one angle.

15. One side, the altitude upon that side, and one angle.

Construct an isosceles trapezoid, having given:

16. The bases and one angle.

18. The bases and diagonals.

17. Two adjacent sides and one angle.

19. The bases and the altitude.

20. The bases and the non-parallel sides. Construct a trapezoid, having given:

21. The bases equal to c and d, and the non-parallel sides equal to a and b. (Fig. 1.)

Ex. 29, page 135

22. The bases equal to a and b, and the diagonals equal to c and d.

23. The bases equal to c and d, a third side equal to a, and the altitude equal to b.

24. The bases equal to a and b, one diagonal equal to d, and one angle between the diagonals equal to x. (Fig. 2.)

25. The bases equal to a and b and the angles at the extremities of a equal to Zz and ≤r.

Construct a quadrilateral, having given:

FIG. 2

26. The four sides equal to a, b, c, and d, and an angle equal to 60°. 27. Three sides equal to a, b, and c; the angle formed by a and b equal to 105°; and the angle formed by b and c equal to 60°.

1. Construct a circle tangent to a given semicircle ACD, and also tangent to the diameter AD. (Fig. 1.)

2. If EFHK is a parallelogram, and AK=KH, prove that EK bisects AF. (Fig. 2.)

3. Show how to construct circle O tangent to circle D, and also tangent to AB and AC. AD bisects A. (Fig. 3.)

4. Tracks at street intersections sometimes pass around arcs of circles tangent to the straight track. Show how the center of the arcs is found. (Fig. 4.)

B

5. Describe circles about the vertices of a triangle so that each shall that be tangent to the other two.

(Fig. 5.)

FIG. 4

6. Construct a circle which shall be tangent to the sides of a given angle, HBK, and pass through a given point A on the bisector of the angle. (Fig. 6.) SUGGESTION. DE 1 AB. DO bisects ZHDA.

7. In locating and drilling holes, a machinist must find the diameters of three circular disks that, placed tangent to each other as shown in Fig. 7, will have their centers 0.765 in., 0.710 in., and 0.850 in. apart. Find the diameters of the disks.

Show that however placed each tangent to the other two the distances between centers of disks will be the same.

8. A given line AC, one inch long, moves so that its ends always touch two given perpendicular lines EF and BD, making right triangles. Where are all points through which the middle point K of the hypotenuse, passes? (Fig. 8.)

E

A

B

A

D

FIG. 8

F

9. Moldings are very important architectural features. They are used to improve the appearance of the angles and projections in panels, cornices, arches, doors, and windows of buildings. The shape of moldings varies according to the style of architecture. Roman moldings are formed from arcs of circles and straight lines. Some of these forms are shown in the figures. Draw them on a larger scale and explain.

10. Construct the following window designs. ABDO is equilateral.

B

11. Show how the following steel ceiling designs are formed.

12. In a given equilateral triangle ABC (Fig. 1), construct three circles, each tangent to two sides of the triangle, and to each of the other two circles.

SUGGESTION. BF bisects ZB, and FE bisects ZBFA. EK is the radius of the three circles required.

1. State theorems concerning angles formed by radii; chords; secants; tangents; chords and tangents.

2. State as many facts as possible about the perpendicular bisector of a chord.

3. What facts can be stated about the angles of an inscribed quadrilateral? Of a circumscribed quadrilateral?

4. Give as many ways as you can for drawing equal angles. Parallel lines.

5. Name as many sets of conditions as you can that will determine a circle.

6. Given two tangents to a circle and two radii drawn to the points of tangency. State the relations of the angles between the tangents and between the radii.

7. State as many loci problems as you can that involve chords in a circle. That involve tangent circles.

8. Give several ways for bisecting an arc. Can an arc be trisected by geometric methods? What are geometric methods? Can a straight line be trisected by geometric methods?

9. Show how to construct a right triangle that has a given hypotenuse. 10. Name several sets of given parts that will determine a triangle uniquely.

11. Make a list of the most useful facts of the present chapter.

CHAPTER III

AREAS OF POLYGONS

MEASURING

332. Definitions. To measure a quantity of any kind is to find how many times it contains a unit of measure.

The number of times a quantity contains the unit of measure is called the numerical measure of that quantity.

The unit of measure may be any convenient quantity of the same kind as that which is to be measured. Certain units are determined by law and are called standard units. In the United States the meter is the standard unit of length from which all other units of length are determined. A convenient unit for measuring the length of the school room is a foot or a meter; for measuring the length of a book, the inch or the centimeter is convenient. The machinist often uses a thousandth of an inch as a unit.

333. Every geometric magnitude has a numerical measure.

The numerical measure may be an integer, a fraction, or an irrational number. For example, a segment of a line may have a length of 5 inches, 5 inches, or √2 inches.

334. Theoretically, the numerical measure, or, simply the measure of a quantity, means its exact measurement. Practically, a quantity can be measured only approximately.

Thus, one cannot say that a board is exactly 10 feet long. It may be a fraction of an inch more or less than 10 feet. The diameter of a steel rod may be given as 0.457 inch, but it may differ from this by a few tenthousandths of an inch.

335. Commensurable and Incommensurable Quantities. If two quantities are such that a common unit of measure is contained in each of them an integral number of times, they are said to be commensurable. If two quantities are such that