Hannah’s favorite toy character is sold wearing one of many outfits. an outfit consists of one shirt, one pair of pants, and one cap. the shirt can be striped, plain, or polka-dotted. the pants may be denim or plaid. the cap may be a cowboy hat or a baseball cap. if hannah picks a toy at random from the shelf and there are an equal number of toys wearing each outfit, what is the probability that the toy is dressed in a polka-dotted shirt and a cowboy hat?

the answer is 2 out of 6

or 2/6

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.