593. What is the radius of a circle whose area is 6 sq. ft.? 594. What is the radius of a circle whose circumference is 12 in.? 595. What is the radius of a circle whose area is n times that of a circle with radius R ? 596. What is the area of the ring between two concentric circles whose radii are 5 ft. and 7 ft. respectively? 597. Within a circle whose radius is R, a circle is drawn so as to cover of the surface of the first circle. What is the radius of the second circle? 598. The radii of two similar segments are as 4 to 7; if the first segment contains 25 sq. in., what does the other contain ? 599. In a white circle of 3 in. radius, an inscribed square is painted black. How much white surface will remain ? 600. In a white square whose side is 4 in., an inscribed circle is painted red. How much white surface will remain ? 601. If the radius of a circle is 10 in., what is the side of the inscribed equilateral triangle? 602. If the side of an inscribed equilateral triangle is 10 in., what is the radius of the circle? 603. If the radius of a circle is 8 in., what is the side of a regular inscribed pentagon ? 604. If the side of a regular inscribed pentagon is 9 in., what is the radius of the circle ? 605. If the radius of a circle is 6 in., what is the side of a regular inscribed octagon ? 606. What must be the radius of a circle so that a side of a regular inscribed octagon shall be 10 in. ? 607. If the radius of a circle is 10 in., what is the side of a regular inscribed decagon ? 608. What must be the radius of a circle so that a side of a regular inscribed decagon shall be 3 in. ? THEOREMS. 609. An angle of a regular polygon of n sides is to an angle of a regular polygon of n + 2 sides, as n2 - 4 is to n2. 610. If the bisectors of all the angles of a polygon meet in a point, a circle can be inscribed in that polygon. 611. An inscribed equilateral triangle is equivalent to half the regular hexagon of the same radius. 612. 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. 613. The altitude of an equilateral triangle is to the radius of the circumscribing circle as 3 is to 2. 614. The area of the regular hexagon is a mean proportional between the areas of the inscribed and circumscribed equilateral triangles. 615. The square of a side of an inscribed equilateral triangle is equivalent to three times the square of a regular hexagon inscribed in the same circle. 616. If the arcs subtended by two sides of an equilateral triangle be bisected, the chord joining those points will be trisected by those sides. 617. The diagonals drawn from a vertex of a regular pentagon to the opposite vertices, trisect that angle. 618. The diagonals drawn from a vertex of any regular polygon of n sides to the opposite vertices, divide the angle into n — 2 equal parts. 619. The diagonals joining alternate vertices of a regular pentagon form by their intercepts a regular pentagon. 620. If the alternate sides of a regular pentagon be produced to meet, the points of meeting will be the vertices of another regular pentagon. 621. The intersecting diagonals of a regular pentagon divide each other in extreme and mean ratio. 622. In a regular pentagon ABCDE, diagonals AC, BE, are drawn, intersecting in F; show that FD is a parallelogram. 623. A ribbon is folded into a flat knot of five edges; show that these edges form a regular pentagon. 624. If P, H, and D denote respectively a side of a regular inscribed pentagon, hexagon, and decagon; then P2H2 + D2. 625. If from any point within a regular polygon of n sides, perpendiculars be drawn to the sides, the sum of these perpendiculars will be equal to n times the apothem. 626. The radius of an inscribed regular polygon is a mean proportional between its apothem and the radius of the similar circumscribed polygon. 627. The area of a circular ring, i.e., the space between two concentric circumferences, is equal to that of a circle having for diameter a chord of the outer circle tangent to the inner circle. 628. If, on the hypotenuse and the arms of a right triangle as diameters, semicircles be described, the curvilinear figures bounded by the greater and the two lesser semicircumferences will be equivalent to the triangle. 637. Describe a square about a given rectangle. 638. Inscribe an equilateral triangle in a given square, so as to have a vertex of the triangle at a vertex of the square. 639. Construct an equilateral triangle that shall be double the area of a given equilateral triangle. 640. Construct a square that shall be of a given square. 641. Construct a regular pentagon that shall be of a given regular pentagon. 642. Construct a regular hexagon that shall be of a given regular hexagon. 643. Describe a circle equivalent to of a given circle. PART II. SOLID GEOMETRY. BOOK VII. PLANES AND POLYHEDRAL ANGLES. PLANES AND PERPENDICULARS. THUS far have been investigated the properties of figures confined to one plane. We are now prepared to enter upon the properties of figures in space; hence the name, Geometry of Space, often applied to this department of the subject, also called Solid Geometry, and Geometry of Three Dimensions. It is to be remembered that, although in our diagrams we can represent only a limited portion of a plane, the plane thus represented is to be regarded as having indefinite extension. PROPOSITION I. THEOREM. 421. Through any given straight line an infinite number of planes can be passed. For if we take any two points a, b, in a plane MN, the straight line drawn joining those points will be wholly in that plane (9). By making the line ab, thus drawn, to coincide with any given line AB, we have one plane MN passing through AB. By revolving MN round AB as an axis, MN can be made to occupy any number of positions, each of which is the position of a plane passing through AB. SCHOLIUM. Hence a plane is not determined that is, marked off from other planes —by the single condition that it passes through a given straight line. To determine a plane, some other condition must be given. PROPOSITION II. THEOREM. 422. A plane is determined by a straight line and a point without that line. M B Given: A straight line AB, and a point not in AB; To Prove: Only one plane can pass through AB and also through C. A plane MN being passed through, and revolved about, AB (421) will have a determined position when it comes to contain the point C. For if it be then turned in either direction about AB, it will cease to contain the point C. Q.E.D. 423. COR. 1. A plane is determined by two straight lines, intersecting or parallel. For it will be determined by either of those lines and any point without it in the other line (422). |