424. COR. 2. A plane is determined by any three points not in the same straight line. For any two of the points may be joined by a straight line, which, with the third point, determines the plane (422). 425. COR. 3. Through a given point in space can pass only one parallel to a given straight line. For in the one plane determined by the line and the point, there can pass through that point only one parallel to the given line (105). 426. COR. 4. The intersection of two planes is a straight line. For their intersection cannot contain three points that are not in the same straight line, seeing that one plane and only one can pass through three such points (424). 427. DEFINITIONS. A straight line is perpendicular to a plane, if it is perpen dicular to every straight line drawn in the plane through its foot, the point in which it meets the plane. The plane, in this case, is also said to be perpendicular to the line. EXERCISE 644. In the diagram for Prop. II., if a plane were passed through AC, and turned about that line as an axis, when would the revolving plane coincide with plane MN? If the plane continued to revolve, when would it again coincide with MN? 645. The intersection of a plane with a surface on which no straight line can be drawn, e.g. the surface of an eggshell or of an apple, is what sort of a line? 646. Enunciate the converse of Cor. 4 of Prop. II.; is that converse generally true? 647. If a circumference be divided into six equal parts, and chords be drawn subtending these parts, two and two, as in the accompanying diagram, is the six-pointed figure thus formed a stellate polygon according to the definition given in Art. 411? If not, why not? PROPOSITION III. THEOREM. 428. A straight line perpendicular to each of two straight lines at their intersection, is perpendicular to the plane of those lines. Given: AP perpendicular to both BB' and CC' in plane MN, at P; To Prove: AP is perpendicular to MN at P. Through P draw any straight line PD in MN; and through D draw BDC so that DB = DC (219). Join A with B, D, and C. Since AD, PD, are medians of ▲ ABC, PBC, resp., (Const.) AB+ AC2 AD2 + 2 BD2, (353) and PB2 + PC2 ≈ 2 PD2 + 2 BD2; .. AB PB+ AC- PC2 AD-2 PD. (Ax. 3) But APB and APC are rt. A; (Hyp.) (being to any line in MN through P.) PROPOSITION IV. THEOREM. 429. A perpendicular to a plane can be drawn from any given point without or in that plane. 1o. Given: A point A without a plane MN; In the plane passed through ▲ and any line BC in MN, draw AD to BC. From D draw, in the plane MN, DE to BC; and, in the plane of AD, DE, draw AP 1 to DE. Then AP is to MN. For since ADC, PDC, and APD are rt. A, (Const.) To Prove: A perpendicular to MN can be drawn at P. From P draw PD to BC, any line in MN; and in any plane intersecting MN in BC, draw DA to BC. Then in to PD. PA is to MN, the plane of PD, DA, draw PA 430. COR. 1. There can be drawn but one perpendicular to a given plane from a point without the plane. N For if there could be two perpendiculars, AP, AQ, from A to MN, draw PQ. Then, in the plane APQ, there would be two perpendiculars from A to the same line PQ, which is impossi ble (51). 431. COR. 2. At a given point in a plane there can be drawn but one perpendicular to that plane. M B A V N Q... For if there could be two perpendiculars, PA, PB, from P in MN, suppose a plane to pass through PA, PB, intersecting MN in PQ. Then from the same point in PQ, and in the same plane, there would be two perpendiculars to PQ, which is impossible (41). 432. COR. 3. The perpendicular is the shortest line that can be drawn to a plane from a given point without the plane. For AP is shorter than any oblique line, AQ, drawn from A to PQ, in the same plane APQ (93). 433. DEFINITION. The distance of a point from a plane is the length of the perpendicular from the point to the plane. EXERCISE 648. In the diagram for Prop. III., the planes determined by AB and P, AC and P, AD and P, intersect in what line? 649. In those planes, if circumferences be described on AB, AC, and AD, as diameters, they will pass through what point and have what common chord ? 650. In the diagram for Prop. V., if AB and AD be joined, and PD = AB, show that (AD + AP) (AD – AP) = AB2. PROPOSITION V. THEOREM. 434. All perpendiculars drawn to a given straight line at a given point, lie in a plane perpendicular to the line. Given: PB, PC, PD, any perpendiculars to AP at P; Through PB, PC, pass the plane MN. Since AP is to PB and PC, (Hyp.) AP is to MN, and MN is to AP. (428, 427) Through AP, PD, suppose a plane passed cutting MN in PD'.' Since AP is to MN, AP is to PD', a line in MN. But AP is to PD; (427) (Hyp.) .. PD must coincide with PD' and lie in MN, Q.E.D. (since, in plane AP, PD, there can be but one to AP at P.) (41) 435. COR. 1. If an indefinite line PB be made to revolve so as to remain always perpendicular to AP at P, it will generate the plane MN at right angles to AP. 436. COR. 2. Through a given point in or without a straight line can be passed but one plane perpendicular to the line. |