400. The area of a parallelogram is equal to the product of its base by its altitude. 401. Parallelograms having equal bases and equal altitudes are equivalent. 403. The area of a triangle is equal to half the product of its base by its altitude. 405. Triangles having equal bases are to each other as their altitudes; triangles having equal altitudes are to each other as their bases; any two triangles are to each other as the products of their bases by their altitudes. 410. The areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles. 412. The areas of two similar polygons are to each other as the squares of any two homologous sides. 414. The homologous sides of two similar polygons have the same ratio as the square roots of their areas. 454. The circumference of a circle is the limit which the perimeters of regular inscribed polygons and of similar circumscribed polygons approach, if the number of sides of the polygons is indefinitely increased; and the area of a circle is the limit which the areas of these polygons approach. 463. The area of a circle is equal to π times the square of its radius. SOLID GEOMETRY. BOOK VI. LINES AND PLANES IN SPACE. DEFINITIONS. 492. DEF. A plane is a surface such that a straight line joining any two points in it lies wholly in the surface. A plane is understood to be indefinite in extent; but is usually represented by a parallelogram lying in the plane. 493. DEF. A plane is said to be determined by given lines or points, if no other plane can contain the given lines or points without coinciding with that plane. 494. COR. 1. One straight line does not determine a plane. For a plane can be made to turn about any straight line AB in it, and thus assume as many different positions as we please. A M N -B 495. COR. 2. A straight line and a point not in the line determine a plane. For, if a plane containing a straight line AB and any point C not in AB is made to revolve either way about AB, it will no longer contain the point C. 496. COR. 3. Three points not in a straight line determine a plane. For by joining two of the points we have a straight line and a point without it, and these determine the plane. $ 495 497. COR. 4. Two intersecting lines determine a plane. For the plane containing one of these lines and any point of the other line not the point of intersection is determined. $ 495 M B N 498. COR. 5. Two parallel lines determine a plane. For two parallel lines lie in a plane (§ 103), and a plane containing either parallel and a point in the other is determined. 499. DEF. When we suppose a plane to be drawn through given points or lines, we are said to pass the plane through the given points or lines. 500. DEF. When a straight line is drawn from a point to a plane, its intersection with the plane is called its foot. 501. DEF. A straight line is perpendicular to a plane, if it is perpendicular to every straight line drawn through its foot in the plane; and the plane is perpendicular to the line. 502. DEF. A straight line and a plane are parallel if they cannot meet, however far both are produced. 503. DEF. A straight line neither perpendicular nor parallel to a plane is said to be oblique to the plane. 504. DEF. Two planes are parallel if they cannot meet, however far they are produced. 505. DEF. The intersection of two planes contains all the points common to the two planes. LINES AND PLANES. PROPOSITION I. THEOREM. 506. If two planes cut each other, their intersection is a straight line. Let MN and PQ be two planes which cut one another. To prove that their intersection is a straight line. Then the straight line AB lies in both planes. § 492 No point not in the line AB can be in both planes; for one plane, and only one, can contain a straight line and a point without the line. § 495 Therefore, the straight line through A and B contains all the points common to the two planes, and is consequently the intersection of the planes. § 505 Q. E. D. PROPOSITION II. THEOREM. 507. If a straight line is perpendicular to each of two other straight lines at their point of intersection, it is perpendicular to the plane of the two lines. To prove that AB is to the plane MN of these lines. Proof. Through B draw in MN any other straight line BE, and draw CD cutting BC, BE, BD, at C, E, and D. Prolong AB to F, making BF equal to AB, and join A and F to each of the points C, E, and D. Then BC and BD are each to AF at its middle point. .. AE = FE; and BE is to AF at B. .. AB is to any and hence every line in MN through B. § 150 § 128 § 143 CE, and ACE = ZFCE. § 161 |