PROPOSITION XI. PROBLEM 505. Through a given point, to draw a line perpendicular to a given plane. I H A M I Given plane PQ and point 4. Required. Through A, to draw a line perpendicular to PQ. Construction. In PQ, draw any line BC. Through A construct a plane AML BC intersecting PQ in DH. Through 4 in the plane AM draw AF DH. AF is the required perpendicular. Proof. Through the foot of AF (F in Fig. I and ▲ in Fig. II) in plane PQ draw IK || BC. 506. COR. 1. Through a given point, one and only one perpendicular can be drawn to a given plane. For if there were two perpendiculars passing through the point, their plane would intersect the given plane in a straight line, perpendicular to both perpendiculars. Or, in a plane, there would be two perpendiculars to a given line, through a given point, which is impossible. 507. COR. 2. If two lines are perpendicular to the same plane, these lines are parallel. Hence DC and DC' coincide, and DC || AB. N Q. E. D. 508. DEF. The distance of a point from a plane is the length of the perpendicular from the point to that plane. Ex. 1452. What is the locus of a point having a given distance from a given plane? PROPOSITION XII. THEOREM 509. If, from a point without a plane, oblique lines be drawn to the plane : (1) Any two lines meeting the plane at equal distances from the foot of the perpendicular are equal. (2) Of two lines meeting the plane at unequal distances from the foot of the perpendicular, the more remote is the greater. E A Given EA perpendicular to plane MN. Oblique lines EB, EC, and ED are drawn so that AB AC and AD > AC. HINT. 1. Prove by means of congruent triangles. 2. ED2 = EA2 + AD2, EC2 = EA2 + AC2. 510. COR. 1. Conversely, equal oblique lines drawn from a point to a plane meet the plane at equal distances from the foot of the perpendicular from the point to the plane, and of two unequal oblique lines, the greater one meets the plane at a greater distance from the foot. 511. COR. 2. The locus of a point in space that is equidistant from all the points in the circumference of a circle is a straight line perpendicular to the plane of the circle, and passing through the center of the circle. 512. COR. 3. The perpendicular to a plane is the shortest line that can be drawn to the plane from a point without. 513. COR. 4. Two parallel planes are everywhere equally distant. Ex. 1453. In the diagram of Prop. XII, if ▲ B = ZC, then EB = EC, and AB = AC. Ex. 1454. In the same diagram, if <B><D, then EB<ED, and AB<AD. Ex. 1455. What is the locus of a point equidistant from two parallel planes ? Ex. 1456. From a point P, 3 in. from a plane MN, a line PA is drawn, A being in MN. If PA 5 in., what is the distance of A from = the foot of the perpendicular drawn from P to MN? PROPOSITION XIII. THEOREM 514. Planes perpendicular to the same straight line are parallel to each other. M B P Given planes MN and PQ perpendicular to line AB. To prove MN PQ. HINT. Apply the indirect method and (503). PROPOSITION XIV. THEOREM 515. A straight line perpendicular to one of two parallel planes is perpendicular to the other also. Proof. Through AB pass any plane intersecting MN in AC and PQ in BD, respectively. Therefore AB is perpendicular to any line in PQ passing through B. Whence AB plane PQ. Q. E. D. 516. COR. Through a given point, one plane and only one can be passed parallel to a given plane. Ex. 1457. If two planes are parallel to a third plane, they are parallel to each other. HINT. Construct a line one of the planes. DIHEDRAL ANGLES 517. If a plane ABC rotates about the line BC until it reaches the position FBC, then the amount of rotation is called a dihedral angle. The line BC is called the edge and the planes ABC and FBC are called the faces of the dihedral angle. NOTE. The student should note that the above statement is not a definition, but merely an explanation, of "dihedral angle." It is impossible to give a definition free from objections. 518. A dihedral angle may be designated by two letters on its edge; or if several dihedral angles have a common edge, by four B E letters, one on each face, and two on the edge, the letters on the edge being placed between the others. Thus, the dihedral angle in the annexed diagram may be designated by BC or ABCE. Obviously the size of the dihedral angle does not depend upon the extent of its faces. Y |