SOLUTION: Parallelogram ABCD is the base of pyramid V-ABCD. DK bisects BC and cuts AC at L. The volume of the pyramid V-ABCD, is what times the volume of pyramid V-LCK?

Question 1198370: Parallelogram ABCD is the base of pyramid V-ABCD. DK bisects BC and cuts AC at L. The volume of the pyramid V-ABCD, is what times the volume of pyramid V-LCK?
Answer by greenestamps(13203) (Show Source): You can put this solution on YOUR website!

Both pyramids have the same height, so the answer depends only on the areas of the bases of the two pyramids. Those bases are parallelogram ABCD and triangle LCK.

Triangles CLK and ALD are similar; and CK is half the length of AD. So the two triangles are similar with a ratio of 1:2.

So the base of triangle CLK is half the base of triangle ALD.

Furthermore, since the ratio of similarity of the two triangles is 1:2, the altitude of triangle CLK is one-third the height of parallelogram ABCD.

So the area of triangle CLK is (1/2)*(1/3) = 1/6 the area of ABCD.

Then, since the heights of the two pyramids are the same, the volume of pyramid V-LCK is 1/6 the volume of pyramid V-ABCD.

ANSWER: The volume of V-ABCD is 6 times the volume of V-LCK.

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