In trapezoid ABCD, AB is parallel to CD, and AB = 10, BC = 9, CD = 22, and DA = 15. Points P and Q are marked on BC so that BP = PQ = QC = 3, and points R and S are marked on DA so that DR = RS = SA = 5. Find the lengths PS and QR.
Answers
•αnswer•
ABCD is a trapezoid where AB||CD (they are parallel). AB = 20, BC = 15, AD = 13. The height of the trapezoid is 12. What is the length of CD?
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Prakash has the right idea. But as you said, there is more than one answer.
If we follow Prakash's method, we draw perpendiculars from A and B down to CD, let's say to points X and Y respectively. We know these 2 perpendiculars have length 12, so AX = 12 and BY = 12.
Looking at triangle AXD, we know AX = 12, and AD = 13, and we know those are 2/3rds of a Pythagorean triple, so DX = 5.
Similarly, looking at triangle BYC, BY = 12, BC = 15, and those are also 2/3rds of a Pythagorean triple, so we know CY = 9.
We also know that XY is the 4th side of rectangle ABYX. AX = BY, so AB = XY. Therefore XY = 20.
CD = DX + XY + CY = 5 + 9 + 20 = 34
HOWEVER!
This answer presumes that CD is the longer side. Which we cannot be sure is the case!
So let's assume that AB is the longer side. Then we draw the perpendiculars from A and B, but we instead end up outside ABCD. So instead, draw them from C and D, since they still count as perpendiculars.
Now we have CY and DX as our perpendiculars, and using the same method as before, we get AX = 5 and BY = 9.
Here's the difference: instead of adding them to the length of AB, we subtract them to find CD. So CD = AB - AX - BY = 20 - 5 - 9 = 6.
So CD = 34 or CD = 6.
QED