Given rectangle abcd with vertices a(1, 4), b(3, 6), c(6, 3), and d(4, 1), what is the
perimeter of abcd? leave your answer in simplified radical form.

  10√2

Step-by-step explanation:

A(1, 4), B(3, 6), C(6, 3), D(4, 1)

The distance formula tells you the distance d between two points (x1, y1) and (x2, y2) is given by ...

  d = √((x2-x1)² +(y2-y1)²

Then the side lengths are ...

  AB = √(2² +2²) = √8 = 2√2

  BC = √(3² +(-3)²) = √18 = 3√2

The perimeter is twice the sum of these sides, so is ...

  P = 2(2√2 +3√2) = 10√2 . . . the perimeter of the rectangle

d. millimeters per hour.

step-by-step explanation:

we are given that,

carlo used a rain gauge to measure the amount of water his lawn is receiving from the sprinklers.

since, according to the options, we see that,

water cannot be measured by 'miles per hour', 'meters per second' and 'grams per seconds'.

we get that, the water can be measured by 'millimeters per hour'.

hence, the appropriate rate for describing the amount of water is 'millimeters per hour'.

∠1≅∠2 by the alternate exterior angles theorem.

step-by-step explanation:

given, a ∥ b and ∠1 ≅ ∠3 .we have to prove that e ∥ f

we know that ∠1≅∠3 and that a || b because they are given. we see that by the alternate exterior angles theorem. therefore, ∠2≅∠3 by the transitive property. so, we can conclude that e || f by the converse alternate exterior angles theorem.

we have to fill the missing statement.

transitivity property states that if a = b and b = c, then a = c.

now, given ∠1≅∠3 and by transitivity property ∠2≅∠3 .

hence, to apply transitivity property one angle must be common which is not in result after applying this property which is ∠1.

the only options in which ∠1 is present are ∠1 and ∠2, ∠1 and ∠4

∠1 and ∠4 is not possible ∵ after applying transitivity we didn't get ∠4.

hence, the missing statement is ∠1≅∠2.

so, ∠1≅∠2 by the alternate exterior angles theorem.