Ex. 718. Find the side of a regular hexagon circumscribed about a circle whose diameter is 1.

Ex. 719. The apothem of an inscribed regular hexagon is equal to one half the side of the inscribed equilateral triangle.

Ex. 720. The area of a ring bounded by two concentric circumferences is equal to the area of a circle whose diameter is a chord of the outer circumference and is tangent to the inner circumference.

Ex. 721. If the radius of a circle is r, find the area of a segment whose chord is one side of a regular inscribed hexagon.

Ex. 722. Three equal circles with a radius of 12 ft. are drawn tangent to each other. What is the area between them?

Ex. 723. The area of an inscribed regular hexagon is equal to three fourths that of a regular hexagon circumscribed about the same circle.

Ex. 724. The altitude of an equilateral triangle is equal to the side of an equilateral triangle inscribed in a circle whose diameter is the base of the first triangle.

Ex. 725. If the radius of a circle is r and the side of a regular inscribed 2 ar polygon is a, prove that the side of a similar circumscribed polygon is

√4r2-a2

Ex. 726. If the alternate vertices of a regular hexagon are joined by straight lines, another regular hexagon is formed which is one third as large as the original hexagon.

Ex. 727. The diagonals of a regular pentagon divide each other in extreme and mean ratio.

PROBLEMS OF CONSTRUCTION

= √ab.

Ex. 728. Construct x, if x =

Ex. 729. Inscribe a circle in a given sector.

Ex. 730. In a given circle describe three equal circles tangent to each other and to the given circle.

Ex. 731. Divide a circle into two segments such that an angle inscribed in one shall be three times an angle inscribed in the other.

Ex. 732. Construct a circumference equal to the sum of two given circumferences.

Ex. 733. Inscribe a square in a given quadrant.

Ex. 734. Inscribe a square in a given segment of a circle.

Ex. 735. Through a given point draw a line so that it shall divide a given circumference into two parts having the ratio 3:7.

Ex. 736. Construct a circle equivalent to twice a given circle.

Ex. 737. Construct a circle equivalent to three times a given circle.

SOLID GEOMETRY

BOOK VII

PLANES AND SOLID ANGLES

427. A plane is a surface such that a straight line joining any two of its points lies wholly in the surface. § 14.

A plane is considered to be indefinite in extent, but in a diagram it is usually represented by a quadrilateral segment.

428. The student will be aided in obtaining correct concepts of the truths presented in the geometry of planes by using pieces of cardboard or paper to represent planes, and drawing such lines upon them as are required. Pins may be used to represent the lines which are perpendicular or oblique to the planes.

429. 1. By using cardboard to represent a plane and the point of a pin or pencil to represent a point in space, discover in how many directions the plane may be passed through the point.

2. By using a card as before and the points of a pair of dividers to represent two fixed points in space, discover whether the number of directions that the plane may take is greater or less than when it was passed through one fixed point.

3. Suppose a plane is passed through three fixed points not in the same straight line, how many directions may it take? How many points, then, determine the position of a plane?

4. Since two of the points must be in a straight line, what else besides three points determine the position of a plane?

5. Since a straight line through the other point may intersect the straight line joining the two points, what else will determine the position of a plane?

6. Since a straight line may join two of the points and a straight line parallel to that may be drawn through the other point, how else may the position of a plane be determined?

In what ways, then, may the position of a plane be determined?

430. A plane is determined by certain points or lines, when it is the only plane which contains those points or lines.

A plane is determined by

1. Three points not in the same straight line.

2. A straight line and a point without that line.

3. Two intersecting straight lines.

4. Two parallel straight lines.

431. The point at which a line meets a plane is called the Foot of the line.

432. A straight line that is perpendicular to every straight line in a plane drawn through its foot is perpendicular to the plane.

In this case the plane is perpendicular to the line.

433. A straight line that is not perpendicular to every line in a plane drawn through its foot is oblique to the plane.

434. A straight line and a plane which cannot meet, however far they may be produced, are parallel to each other.

435. Two planes which cannot meet, however far they may be produced, are parallel to each other.

436. The locus of the points common to two non-parallel planes is the Intersection of the planes.

437. The foot of the perpendicular, let fall from a point to a plane, is called the Projection of the point on the plane.

438. The locus of the projections on a plane of all points in a line is called the Projection of the line.

439. The angle which a straight line makes with a plane is the

440. The distance from a point to a plane is understood to be the perpendicular distance from that point to the plane.

Proposition I

441. Place two planes * so that they intersect. What kind of a line is the line of their intersection?

Theorem. The intersection of two planes is a straight

line.

Data: Any two intersecting planes,

as MN and PQ.

To prove the intersection of MN and M PQ a straight line.

E

F

Proof. Suppose that E and F are any two of the points in which IN and PQ intersect. Draw the straight line EF.

Since E and F are points in the plane MN, § 427, the straight line joining them must lie in MN; and since they are also points in PQ, the straight line joining them must lie in PQ.

Hence,

EF is common to MN and PQ;

EF is a straight line;

that is, § 436, EF is the intersection of MN and PQ.

But, const.,

hence, the intersection of MN and PQ is a straight line.

Q.E.D.

* The student may represent planes and lines as suggested in § 428.

SOLID GEOMETRY.-BOOK VII.

Proposition II

lane draw two intersecting straight lines. If a third perpendicular to each of these at their point of intersecdirection with reference to the plane?

If a straight line is perpendicular to each of aight lines at their point of intersection, it is r to the plane of the two lines.

rough E, in the plane MN, draw any other straight also draw 4C intersecting JK in L.

Ethrough MN to F, making EF =

HE, and draw HA,

ly, HE is perpendicular to every straight line drawn

h E.

32, HE is perpendicular to MN.

etc.

Q.E.D.

A straight line, which is perpendicular to a plane at