13. Given an equilateral triangle inscribed in a circle, and a similar circumscribed triangle; determine the ratio of the two triangles to each other.

14. The diameter of a circle is 20 feet; find the area of a sector whose arc is 120°.

15. The circumference of a circle is 200 feet; find its

area.

16. The area of a circle is 78.54 square yards; find

its diameter.

17. The radius of a circle is 10 feet, and the area of a circular sector 100 square feet; find the arc of the sector in degrees.

18. Show that the area of an equilateral triangle circumscribed about a circle is greater than that of a square circumscribed about the same circle.

19. Let AC and BD be diameters perpendicular to each other; from P, the middle point of the radius OA, as a centre, and a radius equal to PB, describe an are cutting OC in Q; show that the radius OC is divided in extreme and mean ratio at Q.

B

20. Show that the square of the side of a regular inscribed pentagon is equal to the square of the side of a regular inscribed decagon increased by the square of the radius of the circumscribing circle.

21. Show how, from 19 and 20, to inscribe a regular pentagon in a given circle.

22. The side of a regular pentagon, inscribed in a circle, is 5 feet, and that of a regular inscribed decagon is 2.65 feet; find the side and the area of a regular hexagon inscribed in the same circle.

BOOK VI.

PLANES AND POLYEDRAL

ANGLES.

DEFINITIONS.

1. A straight line is PERPENDICULAR TO A PLANE, when it is perpendicular to every straight line of the plane which passes through its FOOT; that is, through the point in which it meets the plane.

In this case, the plane is also perpendicular to the line.

2. A straight line is PARALLEL TO can not meet the plane, how far soever both may be proA PLANE, when it

duced.

In this case, the plane is also parallel to the line.

3. TWO PLANES ARE PARALLEL, when they can not meet, how far soever both may be produced.

4. A DIEDRAL ANGLE is the amount of divergence of two planes.

The line in which the planes meet is called the edge of the angle, and the planes themselves are called faces of the angle.

The measure of a diedral angle is the same as that of a plane angle formed by two straight lines, one in each face, and both perpendicular to the edge at the same point. A diedral angle may be acute, obtuse, or a right angle. In the latter case, the faces are perpendicular to each other.

5. A POLYEDRAL ANGLE is the amount of divergence of several planes meeting at a common point.

This point is called the vertex of the angle; the lines in which the planes meet are called edges of the angle, and the portions of the planes lying between the edges are called faces of the angle. Thus, S is the vertex of the polyedral angle, whose edges are SA, SB, SC, SD, and whose faces are ASB, BSC, CSD, DSA.

A polyedral angle which has but three faces, is called a

triedral

B

S

angle.

POSTULATE.

A straight line may be drawn perpendicular to a plane from any point of the plane, or from any point without the plane.

If a straight line has two of its points in a plane, it lies wholly in that plane.

For, by definition, a plane is a surface such, that if any two of its points are joined by a straight line, that line lies wholly in the surface (B. I., D. 8).

Cor. Through any point of a plane, an infinite number of straight lines may be drawn which lie in the plane. For, if a straight line is drawn from the given point to any other point of the plane, that line lies wholly in the plane.

Scholium. If any two points of a plane are joined by a straight line, the plane may be turned about that line as

an axis, so as to take an infinite number of positions. Hence, we infer that an infinite number of planes may be passed through a given straight line.

PROPOSITION II. THEOREM.

Through three points, not in the same straight line, one plane can be passed, and only one.

Let A, B, and C be the three points: then can plane be passed through them, and only one.

Join two of the points, as A and B, by the line AB. Through AB let a plane be passed, and let this plane be turned around AB until it contains the point C; in this position it will pass through the

one

three points A, B, and C. If now, the plane be turned about AB, in either direction, it will no longer contain the point C: hence, one plane can always be passed through three points, and only one; which was to be proved.

Cor. 1. Three points, not in a straight line, determine the position of a plane, because only one plane can be passed through them.

Cor. 2. A straight line and a point without that line determine the position of a plane, because only one plane can be passed through them.

Cor. 3. Two straight lines which intersect determine the position of a plane. For, let AB and AC intersect at A: then either line, as AB, and one point of the other, as C, determine the position of a plane.

Cor. 4. Two parallel straight lines determine the position

A

of a plane. For, let AB and CD be parallel. By definition (B. I., D. 16) two parallel lines always lie in the same plane. But either line, as AB, and any point of the other, as F, determine the position of a plane: hence, two parallels determine the position of a plane.

PROPOSITION III. THEOREM.

The intersection of two planes is a straight line.

Let AB and CD be two planes: then is their intersection a straight line.

For, let E and F be any two points common to the planes; draw the straight line

EF.

B

This line having two points in the plane AB, lies wholly in that plane; and having two points in the plane CD, lies wholly in that plane: hence, every point of EF is common to both planes. Furthermore, the planes can have no common point lying without EF, otherwise there would be two planes passing through a straight line and a point lying without it, which is impossible (P. II., C. 2); hence, the intersection of the two planes is a straight line; which was to be proved.

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

Let MN be the plane of the two lines BB, CC, and let AP be perpendicular to these lines at P: then is AP per