Simplify each expression for the perimeter of the deck

that you wrote in problems 1 and 2. What do you notice?

9514 1404 393

  the simplified expressions are the same

Step-by-step explanation:

(d+5)+(d+5)+(d+5)+(d+5) simplifies to 4d +20

4(d +5) simplifies to 4d +20

We notice that the equivalent expressions simplify to the same expression.

3. parallelogram

step-by-step explanation:

we turn each of our sides into direction vectors

ab=b-a=(3 - 7, 1 - 0) = (-4, 1)

bc=c-b=(1,6)

cd=d-c=(4,-1)

da=a-d=(-1,-6)

we see how the opposite sides have negated direction vectors; this is characteristic of a parallelogram.   we have congruent sides ab=cd and bc=da with unequal lengths because 1^2+6^2 is different from (-4)^2+1^2.   so it's not a square or a rhombus.  

we check the dot product of direction vectors of adjacent sides.   we have

ab dot bc = -4(1) + 1(6) = 2

that's not zero so the sides are not perpendicular.   we've ruled out rectangle.

the direction vectors indicate two pairs of congruent parallel sides, so we choose parallelogram over trapezoid.