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Since the two vertical lines are in the same plane with ab, conceive through b a line to be drawn parallel to ab, and terminated at the other vertical, a right-angled triangle will be formed, of which the base is = ab, and the altitude the difference between the verticals, and the hypothenuse is the distance required, AB.

To construct this triangle, through b' draw I'm parallel to xy, meeting ra' in m, take ml = ab, draw a'l'; then a'l is the distance required.

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PROBLEM 6.

Given the projections of a straight line and a point, to find

those of a line passing through the point, and parallel to the given line.

20. Let ab a'b' be the projections of the line, cc' of the point; through c draw cd parallel to a'b'; through c draw c' d' parallel to a'b': then cd and c'd' are the projections required.

For when two lines are parallel, their projecting planes are also parallel, and :. the intersections of these planes with a third plane are also parallel ; i. e., the projections of two parallel lines are also parallel.

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PROBLEM 7.

Given the traces of a plane, and also one of the projections of

a line in the plane, to find the other projection. 21. Suppose the horizontal projection of the line to be given ; through it suppose a vertical plane to pass, its intersection with the given plane will be the line whose vertical trace is required.

Let aaa' be the given plane, bc the given projection. Draw I to wy, it will be the vertical trace of the plane through cb; and b and c' where the traces of the two planes intersect, are points of the line which has bc for ts horizontal projection. Draw :: bb'

Ito xy, join b'd', it will be the vertical projection required.

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PROBLEM 8.

Through a given point, to draw a plane parallel to a given

plane.

22. Let a, a' be the projections of the given point ; Bb BB', the traces of the given plane. Then, since the intersections of two parallel planes with a third plane are also parallel, therefore the traces of the required plane must be parallel to those of the given plane.

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Through the given point a, a', and in the required plane, suppose a line to be drawn parallel to the horizontal trace of the plane; it will :: be parallel to Bb; its horizontal projection ad must pass through a, and be parallel to Bb;

its vertical projection must pass through a', and be parallel

to xy.

Let ď' be the vertical trace of this line, it is :: a point in the vertical trace of the required plane ; draw :: d'c' parallel to BB', then d'c' is the vertical trace of the required plane: produce c'd to meet sy in y, draw yc parallel to Bb, it will be the horizontal trace of the required line.

To verify this construction, suppose through the given point a line to be drawn parallel to BB', of which the projections are ae and a'é'; the point e, where it meets the horizontal plane, must be on the horizontal trace of the required plane.

PROBLEM 9.

To draw a plane through three given points.

23. Since the points are given, their projections are also given ; let these be a, a, b, b, c and d. Join therefore the points, taken two and two, by the lines ab, a'b, bc, b'd',

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ac, a'c'; which, having each of them two points in the required plane, will be wholly contained in that plane. Find, therefore, the three points of intersection of these lines with the planes of projection, and we shall have the two traces of the required plane; the three points of each trace must be in the same straight line, and the two traces must intersect the ground line in the same point.

Thus let the line ab, a'b' meet the planes of projection in h, and o; the line ac, a'c', in h, and 0,; the line bc, b'c', in h, and 0; consequently the horizontal trace of the required plane will pass through the points h, h', he, and the vertical through 0, 0, 0, and these two traces shall cut xy in the same point t.

Cor. If the line joining two of the points be parallel to one of the planes of projection, as, for instance, the vertical plane, the preceding construction could not be executed. But, in that case, the vertical trace of the required plane is parallel to that line ; or, what is the same thing, to its vertical projection.

PROBLEM 10.

To draw a plane through a given point and through a given

line.

24. Let a and a' be the projections of the given point; bc and b’ď those of the given line; ad and a'd' those of a

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line through the point and parallel to the given line, and therefore situated in the required plane.

Let c and e be the horizontal, and b' and d' the vertical, traces of these lines ; :the lines ce, b'd', are the traces of the required plane, and produced must meet xy in the same point.

PROBLEM 11.

Todraw a plane through a given line parallel to a given

line.

25. In the first line ab, a'b', take any point, f, f', and through it draw a line parallel to the second line cd, c'd'.

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The required plane must contain this line, and the line ab, a'b'; and the traces of these lines will determine the traces of the required plane. (Prob. 9, 10.)

PROBLEM 12.

Through a given point to draw a plane parallel to two given

lines.

26. Through the given point draw lines parallel to the given lines; the intersections of these lines with the planes of

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