Respuesta :

chrysalism
To solve this, you must make use of distance formula, which is defined below:

[tex] \sqrt{(x1 - x2)^2 + (y2 - y1)^2} [/tex]

If we can determine the length and width, we can solve the perimeter. So let's begin! Let's take our first two points, (-1, 4) and (5, 2) and plug the values in.

[tex] \sqrt{(5 - (-1))^2 + (2 - 4)^2} [/tex]

That's a bit of a mess, so let's simplify. By evaluating(which will take too long to work out on here), we get the square root of 40.

Now for the next set of points: (5, 2) and (4, -1). When we plug them in and evaluate the distance formula, we get the square root of ten.

Now that we have two values, we can simply multiply each by two and add them, because they represent side lengths.

If you plug this entire thing into a calculator, it gives you something like 18.97, which can be rounded up to 19.0 as our answer. So your choice was correct.

Hope this is helpful! :)
Beatles11
First, I would suggest for you to sketch out this rectangle, this will help you choose the pairs with which you will calculate the side lengths.

Then, let's calculate the difference between the two points which are the closest together. (The short side of the rectangle) You can choose between (-2,1) and (-1,4) OR (4,-1) and (5,2). Here, I chose the first pair:

Also, here's the formula to find the distance between two points:

d (distance) = √(x(#2)−x(#1))²+ (y(#2)−y(#1))²

(I know this seems really complicated, but it actually isn't. I tried my best to make it easy for you to understand, but there are some special numbers I can't make with Brainly's keyboard. I highly recommend that you look up the formula. it will be much clearer. ) 

Distance between (-1,4) and (-2, 1) = √ ( -2 - (-1))² + (1 - 4)²
                                                         =  √ (-1)² + (-3)²
                                                         =  √ 1 + 9
                                                         = √ 10

So now that we know the length of the short sides, we can just double it to have the length of both short sides of the rectangle:

Short side + Short side = √10 + √10
                                       = 2√10

Now, we will calculate the distance between the two points which are the farthest from each other! (The long side of the rectangle) You can choose either (-1,4) and (5,2) OR (-2,1) and (4,-1). Here I chose the first pair:
 
Distance between (-1,4) and (5,2) = √ ( 5 - (-1))² + (2 - 4)²
                                                        = √ (6)² + (-2)²
                                                        = √ 36 + 4
                                                        = √ 40
                                                        = 2 √ 10

And now, we can calculate the lengths of both long sides of the rectangle:

Long side + Long side = 2 √ 10  + 2 √ 10
                                      = 4√ 10

So, as you can see, now that we have all the side lengths, we can add them all up and FINALLY get:

2 short sides + 2 long sides = 2 √ 10 + 4 √ 10
                                               = 6 √ 10

Therefore, the perimeter of this rectangle is 6√ 10 units! But in decimal form, that would be 18.97. So as a final answer, (rounded to the nearest tenth of a unit), the perimeter is 19.0 units.

Really hope this helps!
But feel free to message me if you have any questions :)

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