Descriptive geometry . the side b,cr in the point fr. Project from fr perpen-dicular to VX to f° in bc. and from / perpendicular to GLto/* in 6*c*; then evf and ehfh are the projections required. In Fig. 230, the interior angle at h is bisected, the actualbisector, bhf, intersecting drer in the point f. From fr areobtained/* and /, giving bhfh and bfv as the required pro-jections. CHAPTER XVI MISCELLANEOUS PROBLEMS ON THE LINE AND PLANE 143. The Distance from a Point to a Line. The shortest dis-tance from a point to a given straight line is obtained bydropping a perpendicular from the point to
Descriptive geometry . the side b,cr in the point fr. Project from fr perpen-dicular to VX to f° in bc. and from / perpendicular to GLto/* in 6*c*; then evf and ehfh are the projections required. In Fig. 230, the interior angle at h is bisected, the actualbisector, bhf, intersecting drer in the point f. From fr areobtained/* and /, giving bhfh and bfv as the required pro-jections. CHAPTER XVI MISCELLANEOUS PROBLEMS ON THE LINE AND PLANE 143. The Distance from a Point to a Line. The shortest dis-tance from a point to a given straight line is obtained bydropping a perpendicular from the point to the line. Thisperpendicular evidently lies in the plane which is determinedby the given point and line. Problem 25. To find the shortest (perpendicular) distance froma point to a line. Analysis. In the general position of the line, the right anglebetween it and the perpendicular will not project as such(§ 109). Hence, pass an auxiliary plane through the givenline and point. Kevolve this plane into either coordinate. Fig. 232. plane, obtaining thus the true relative position of the line andpoint. Draw the required perpendicular in the revolved posi-tion. To find the projections of the perpendicular, revolvethe auxiliary plane back to its original position. 152 XVI, § 143] PROBLEMS OX THE LIXE AXD PLAXE 153 Construction (Fig. 232). Let A be the given line, and c thegiven point. Through c draw the auxiliary line B {Bh, Bv)parallel to A (§ 93). Pass the auxiliary plane X through thelines A and B (Prob. 6, § 106). Since X is introduced solelyfor the purpose of obtaining a revolved position, but one trace,as HX, is necessary (§ 139). Revolve A and c about HX intoH. To do this most readily, revolve point c to cr (Trob. 21,Working Rule, § 138) ; draw Br through c, and the trace s2(Prob. 21, Corollary, § 138), then draw Ar through the trace sxand parallel to Br. From cr drop the perpendicular cTdr to Ar;this is the actual shortest distance required. To find the pro-jections of
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