So, I am incredibly confused on this question, and I am none the wiser on how to solve it,
A cube whose side length is 5 is shown below. Let P be on edge AB such that AP = 2, and let Q be on edge AE such that AQ = 1. The plane passing through points C, P, and Q intersects DH at R. Find the length DR.
Oh, and if possible, please explain what you did step by step, that would greatly appreciated!
Could you give us a lift into what maths your expected to use ?
There are several methods that could be used.
What are you currently working on ?
Vectors, 3D co-ordinate geometry, what ?
To find the length of DR, we can use the Pythagorean theorem. We know that the length of AC is 5 and the length of AQ is 1. Using the Pythagorean theorem, we can find that the length of CQ is 4. Now we know that CR is a right triangle, we can use the Pythagorean theorem again. We know that the length of CQ is 4 and the length of DR is unknown. Applying the Pythagorean theorem, DR = sqrt(5^2-4^2) = sqrt(9) = 3. so, DR = 3.
Still don't know what method you're expected to use, so I'll just take you through the easiest, (leaving you to fill in the detail).
To begin with though ignore answers #2 and #4, both are incorrect.
#2 is from one of the sites dimwits and #4 is just simply incorrect, beginning with AC = 5, and then ignoring the point P, which is needed to determine the plane CPQ.
Basically, the method is 3D co-ordinate geometry, find the equation of the plane passing through C P and Q, substitute the two known co-ordinates of R, and out pops the third.
So, let A be the origin of the co-ordinate system with AD as the x-axis, AB the y-axis and AE the z-axis. You should then be able to write down the co-ordinates of the points P Q and C. (P will be (0, 2, 0) for example).
The equation of a plane in 3D can be written as ax + by + cz = 1. Substitute the co-ordinates of P Q and C to determine the values of a b and c and then, having done that, substitute the two known co-ordinates of R. The third co-ordinate of R (the one that you need), drops out.
Post your answer if you wish.
If you like the method (and like to play with it), you could repeat it with different co-ordinate axes. For example, let B be the origin with BA BC and BF as the axes, the arithmetic will be different but you should arrive at the same result. Any of the cubes corners could be used as the origin of the co-ordinate system.