SOLUTION: In (square symbol?) ABCD, P, Q, R, and S are mid-points of the sides. The perimeter of (square symbol?) ABCD is 36. AC = 12 and BD = 9. What is the perimeter of (square symbol?) PQ

Question 1012063: In (square symbol?) ABCD, P, Q, R, and S are mid-points of the sides. The perimeter of (square symbol?) ABCD is 36. AC = 12 and BD = 9. What is the perimeter of (square symbol?) PQRS?
Image: https://gyazo.com/fe3ba69d233d918426765ee85b85dc1f
Found 2 solutions by jim_thompson5910, ikleyn:
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
In the figure, ABCD is NOT a square and it is NOT a rectangle. It's simply a quadrilateral. If you look at the information given that AC = 12 and BD = 9, this also tells us we don't have a rectangle (if we don't have a rectangle, it's certainly not a square). The diagonals have to be equal in order to have a rectangle.

However, PQRS is a parallelogram. This is true no matter where you place A,B,C or D. The proof of that is here. We'll use the fact that the opposite sides of any parallelogram are congruent.

Connect A to C. This segment is 12 units long. Once you have segment AC drawn, you have triangle ABC. The segment PQ is parallel to AC and it's half as long. This is midsegment property of triangles.

So,

PQ = (1/2)*(AC)
PQ = (1/2)*12
PQ = 6

So PQ is 6 units long. So is SR (opposite side is congruent).

Using the same idea for BD and PS, we get

PS = (1/2)*(BD)
PS = (1/2)*9
PS = 4.5

The opposite sides of the parallelogram are congruent, so PS = RQ

Now we know all 4 sides of the parallelogram PQRS and they are

PQ = 6
SR = 6
PS = 4.5
RQ = 4.5

Add up the four sides:
PQ+SR+PS+RQ = 6+6+4.5+4.5 = 21

This is no coincidence that the perimeter of PQRS is equal to AC + BD. So in the future, you can use this shortcut to make things faster.
Answer by ikleyn(52814) (Show Source): You can put this solution on YOUR website!
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See also the lesson Midpoints of a quadrilateral are vertices of the parallelogram in this site.

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