### Video Transcript

The coordinates of points ๐ด, ๐ต, and ๐ถ are ๐พ, negative two; two, eight; and
negative nine, six, respectively. Given that line ๐ด๐ต is equal to line ๐ต๐ถ, find all possible values of ๐พ.

First, letโs just sketch a coordinate plane to try and get a sense of whatโs
happening. Point ๐ด we donโt know itโs ๐ฅ-value; itโs missing. But we do know that itโs located at negative two for its ๐ฆ-value. Point ๐ด is located somewhere on this line. Point ๐ต is located at two, eight; point ๐ถ at negative nine, six. And this distance โ the distance from ๐ต to ๐ถ โ will be equal to the same distance
from point ๐ด to point ๐ต.

Sketching this can be helpful, but it wonโt give us an exact answer. To do that, weโll need to use the distance formula. To find the distance between two points, we take the square root of the difference
between the two points ๐ฅ-values and ๐ฆ-values squared.

Weโll start out by finding the distance from point ๐ถ to point ๐ต. ๐ฅ two minus ๐ฅ one squared plus ๐ฆ two minus ๐ฆ one squared. Two minus negative nine equals 11 โ we keep our square, 11 squared โ plus eight minus
six equals two and keep the square. We still need to take the square root of this value. 11 squared equals 121 plus two squared which equals four.

The distance between ๐ต and ๐ถ is the square root of 125. Weโre gonna take this distance and use it to solve the distance formula a second
time. The square root of 125 is the distance from point ๐ด to point ๐ต. Point ๐ด is at ๐พ, negative two. Point ๐ต is at two, eight. We can use point ๐ด as ๐ฅ one, ๐ฆ one and point ๐ต as ๐ฅ two, ๐ฆ two.

Plugging this into our equation, we get two minus ๐พ squared plus eight minus
negative two squared. Two minus ๐พ squared cannot be simplified yet. But in place of eight minus negative two, we can say 10 squared. We notice that weโre taking the square root of both sides of the equation. We can square both sides of our equation.

The square root of 125 squared equals 125. And on the right, it would say two minus ๐พ squared plus 10 squared. 10 squared equals 100. We can subtract 100 from both sides of the equation. 125 minus 100 equals 25. And on the right, two minus ๐พ squared. To get rid of that squared, weโll take the square root of both sides of the
equation.

Hereโs where we really have to pay attention. The square root of 25 is five. But itโs also negative five. We need plus or minus five is equal to two minus ๐พ. We need to break this equation up into two pieces: two minus ๐พ equals positive five
and two minus ๐พ equals negative five.

For the left equation, Iโll subtract two from both sides. This leaves me with negative ๐พ equals three. But Iโm not interested in negative ๐พ. I want to know what positive ๐พ is. And that means Iโll multiply the whole equation by negative one. ๐พ is equal to negative three.

Weโll do the same thing with the right equation, subtract two from both sides. Negative ๐พ is equal to negative seven. But we want positive ๐พ. So weโll multiply by negative one. And ๐พ is equal to seven.

At ๐พ equals negative three, the line ๐ด๐ต has a distance of the square root of
125. Thereโs a possibility that ๐ด could be located at seven, negative two or negative
three, negative two.

The possible values for ๐พ such that line ๐ด๐ต and line ๐ต๐ถ are equal is negative
three or seven.