Question

What’s the answer to this question I’ve been working on it for along time.

Answer 1

Answer:

$4\sqrt{5}$

Step-by-step explanation:

Well if you draw the distance between the two by drawing a vertical line, and a horizontal line to connect them, you'll notice that a right triangle forms. I provided an image to demonstrate this.

So, generally to find the "distance" between two numbers, you would do: $|a-b|$, since "a" might be less than "b" thus resulting in a negative length, so the absolute value makes it positive. The point is that $|a-b| = |b-a|$ the order doesn't matter.

Assuming a doesn't equal b, then either a-b or b-a will be positive, and one will be negative, but their absolute values will be the same.

The reason I'm mentioning this, is because the distance formula is: $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2$

this is derived from the Pythagorean Theorem: $a^2+b^2=c^2$, in which "c" represents the hypotenuse. We can solve for c by taking the square root of both sides to get: $c=\sqrt{a^2+b^2}$

The main difference between the Pythagorean Theorem, and the distance formula, is that you're solving for the side lengths given two points, unlike when you are given a triangle, and you already know the side lengths.

The x_2 - x_1, represents the horizontal length, and the y_2 - y_1 represents the vertical length.

But what if x_2 is less than x_1? The length would become negative right? That is true, but as mentioned before the absolute values are the same, so after squaring the value, if it was negative it becomes positive, and it becomes the same had you done x_1 - x_2.

The main point here is: $|x_2-x_1|=|x_1-x_2|$ and the same goes for y, so that means that: $(x_2-x_1)^2=(x_1-x_2)^2$

Anyways, now we can plug the points into the equation:

$\sqrt{(-4-4)^2+(2-(-2))^2}\\\sqrt{(-8)^2+(4)^2}\\\sqrt{64+16}\\\sqrt{80}$

Now all we have to do here is simplify the radical, by using the radical identity: $\sqrt[n]{a*b}=\sqrt[n]{a}*\sqrt[n]{b}$

We need to look for the greatest factor, that is a perfect square.

By listing the factors we get: 1, 2, 4, 5, 8, 10, 16, 20, 40 and 80.

The greatest factor which is a perfect square is 16, so we can express the radical as: $\sqrt{16*5}\implies\sqrt{16}*\sqrt{5}\implies4*\sqrt{5}$

Answer 2

Answer:

$\sqrt{80}$ or $8.94$.

Step-by-step explanation:

For this problem, let's refer to the distance formula for points in a cartesian plane.

Distance formula:

$D= \sqrt{(x_{2}-{x_{1})^{2} + (y_{2}-{y_{1})^{2} }$

Now, identify the coordinates you have on this plane:

A: (-4, 2)

B: (4, -2)

Now, the coordinates that go with the number 1 is the point that is located closer to -∞, or just the one that has a smaller value of x.

$x_{1} =-4\\\\y_{1} =2\\\\x_{2} =4\\\\y_{y} =-2$

Substitute in the formula and calcualte (check attatched image below).

Important: You cannot assign x₁ or x₂ arbitrary, the coordinates that goes with x₁ or y₁ is always the one that has the smaller value of x. In this case it's the A point, because their x value (-4) is smaller than the x value of B (4).