To solve this problem we need to remember the distance formula for points on a coordinate plane:

\(\displaystyle d=\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}\)

In this case, \(\displaystyle (x_1,y_1)=(-7,4)\) and \(\displaystyle (x_2,y_2)=(-12,16)\)

\(\displaystyle d=\sqrt{((-7)-(-12))^{2}+(4-16)^{2}}\)

\(\displaystyle d=\sqrt{5^{2}+(-12)^{2}}\)

\(\displaystyle d=\sqrt{25+144}=\sqrt{169}=13\)

The question is merely asking the distance between two points. This kind of problem can be quickly solved for by using the distance formula:

\(\displaystyle d= \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\), where \(\displaystyle d\) is distance and \(\displaystyle x_1\), \(\displaystyle x_2\), \(\displaystyle y_1\), and \(\displaystyle y_2\) come from the given points.

This problem merely needs to have the \(\displaystyle x\) and \(\displaystyle y\) values substituted in for so we can solve for \(\displaystyle d\). 

Arbitrarily assigning \(\displaystyle (x_1,y_1)\) and \(\displaystyle (x_2,y_2)\), we substitute in our values as follows:

\(\displaystyle d=\sqrt{(12-7)^2+(0-25)^2}\)

\(\displaystyle d=\sqrt{(5)^2+(-25)^2}\)

\(\displaystyle d=\sqrt{25+625}\)

\(\displaystyle d=\sqrt{650}\)

\(\displaystyle d=25.495 \approx {\color{Blue} 25.5}\)

The question is merely asking the distance between two points. This kind of problem can be quickly solved for by using the distance formula:

\(\displaystyle d= \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\), where \(\displaystyle d\) is distance and \(\displaystyle x_1\), \(\displaystyle x_2\), \(\displaystyle y_1\), and \(\displaystyle y_2\) come from the given points.

This problem merely needs to have the \(\displaystyle x\) and \(\displaystyle y\) values substituted in for so we can solve for \(\displaystyle d\). 

Arbitrarily assigning \(\displaystyle (x_1,y_1)\) and \(\displaystyle (x_2,y_2)\), we substitute in our values as follows:

\(\displaystyle d=\sqrt{(1+9)^2+(3-32)^2}\)

\(\displaystyle d=\sqrt{(10)^2+(-29)^2}\)

\(\displaystyle d=\sqrt{100+841}\)

\(\displaystyle d=\sqrt{941}\)

\(\displaystyle d=30.6757 \approx {\color{Blue} 30.7}\)

What is the distance between the following points?

\(\displaystyle (4,-6)\) \(\displaystyle (-19,-53)\)

To find distance, use distance formula:

\(\displaystyle d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\)

Note, if you cannot recall distance formula think of Pythagorean theorem. When using distance formula, you are simply finding the length of the hypotenuse of a right triangle.

Anyway, start plugging in our points and simplify to the answer. We'll call our first point 1 and our second point 2

\(\displaystyle d=\sqrt{(4--19)^2+(-6--53)^2}=\sqrt{(23)^2+(47)^2}=\sqrt{2738}\approx52.3\)

 So our answer is 52.3

What is the length of the distance between the points \(\displaystyle (9,15)\) and \(\displaystyle (18,3)\)?

Find distance with distance formula, which is quite similar to Pythagorean Theorem

\(\displaystyle d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\)

Now, let's call \(\displaystyle (9,15)\) point 1 and \(\displaystyle (18,3)\) point 2, then let's plug in and find d!

\(\displaystyle d=\sqrt{(9-18)^2+(15-3)^2}=\sqrt{81+144}=\sqrt{225}=15\)

So the distance is 15

Find the distance between the following points:

\(\displaystyle (14,25)(38,7)\)

To find the distance between two points, use distance formula.

\(\displaystyle d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)

Distance formula is closely related to Pythagorean theorem. Pythagorean Theorem is:

\(\displaystyle a^2+b^2=c^2\)

Which can be rewritten as:

\(\displaystyle c=\sqrt{a^2+b^2}\)

Now, in distance formula, we are essentially finding the hypotenuse of a right triangle.

Anyway, to find the distance, we simply need to plug in the points we are given and simplify:

\(\displaystyle d=\sqrt{(38-14)^2+(7-25)^2}=\sqrt{(24^2)+(-18)^2}\)

Continue

\(\displaystyle \sqrt{(24^2)+(-18)^2}=\sqrt{576+324}=\sqrt{900}=30\)

So our hypotenuse (distance) is 30.

If you are really observant, you can see that the other two sides of the triangles are 24 and 18.

Write the distance formula.

\(\displaystyle d= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)

Substitute the values of the points inside the equation.

\(\displaystyle d= \sqrt{(4-1)^2+(2-3)^2}\)

Simplify by order of operations.

\(\displaystyle d= \sqrt{(3)^2+(-1)^2} = \sqrt{9+1} = \sqrt{10}\)

The length of the line is \(\displaystyle \sqrt{10}\).

To find the distance between two points using the distance formula, we use the following formula:

\(\displaystyle d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

where \(\displaystyle (x_1, y_1)\) and \(\displaystyle (x_2, y_2)\) are the given points.  So, we can substitute the points \(\displaystyle (2, 6)\) and \(\displaystyle (-4, 8)\).

\(\displaystyle d = \sqrt{(-4 - 2)^2 + (8 - 6)^2}\)

\(\displaystyle d = \sqrt{(-6)^2 + (2)^2}\)

\(\displaystyle d = \sqrt{36 + 4}\)

\(\displaystyle d = \sqrt{40}\)

Therefore, the distance is \(\displaystyle \sqrt{40}\).

To solve, we will use the distance formula:

\(\displaystyle \text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

where \(\displaystyle (x_1, y_1)\) and \(\displaystyle (x_2, y_2)\) are the given points.  Given the points

\(\displaystyle (-7, 2)\) and \(\displaystyle (5, 9)\)

we can substitute into the formula.  We get,

\(\displaystyle \text{distance} = \sqrt{(5 - - 7)^2 + (9 - 2)^2}\)

\(\displaystyle \text{distance} = \sqrt{(12)^2 + (7)^2}\)

\(\displaystyle \text{distance} = \sqrt{144 + 49}\)

\(\displaystyle \text{distance} = \sqrt{193}\)

Therefore, the length of the line with the endpoints (-7, 2) and (5, 9) is \(\displaystyle \sqrt{193}\).

The distance formula equation is as follows:

\(\displaystyle d = \sqrt{(x1-x2)^{2} + (y1-y2)^{2}}\)   

Note that \(\displaystyle x1,x2,y1, y2\) simply refer to the 'first \(\displaystyle x\) and \(\displaystyle y\)' and 'second \(\displaystyle x\) and \(\displaystyle y\)' points, these are simply for keeping track and do not require any further computation. 

The \(\displaystyle x\) and \(\displaystyle y\) coordinates given can be plugged in to solve for the distance between these two points.

\(\displaystyle d = \sqrt{(8-1)^{2} + (7-5)^{2}}\)

\(\displaystyle d = \sqrt{(7)^{2} + (2)^{2}}\)

\(\displaystyle d = \sqrt{49 + 4}\)

\(\displaystyle d = \sqrt{53}\)

\(\displaystyle d = 7.3\)