Let ABCDEFGH be a cube of side length 5, as shown. Let P and Q be points on line AB and line AE, respectively, such that AP = 3 and AQ = 1. The plane through C, P, and Q intersects line DH at R. Find DR.
Let ABCDEFGH be a cube of side length 5, as shown. Let P and Q be points on line AB and line AE, respectively, such that AP = 3 and AQ = 1. The plane through C, P, and Q intersects line DH at R. Find DR.
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∠CPQ ≈ 68.199º
∠DCR ≅ ∠APQ = 18.43494881º
DR = tan(∠DCR) * CD = 1.66666667 or 5/3
Thanks, jugoslav.....here's another way without using trig.....
Let B = (0,0) A = (5,0) C = (0,5) and D = (5,5) P = (2,0)
Slope of CP = -5/2
Equation of line through CP ..... y = (-5/2)x + 5
Let S be the intersection of the line through segment RQ and the line through CP
The x coordinate of this point = 5....so we have
y = -(5/2)(5) + 5
y= -25/2 + 5
y = -15/2
So S = (5, -15/2)
So SQ = 15/2
Using similar triangles
AQ / SA = DR/SD
AQ/ SA = DR / (SA + AD)
1/ (15/2) = DR / ( 15/2 + 5)
(2/15) = DR / (25/2)
DR = (25/2) (2/15) = 50/30 = 5/3