Video: Finding the Missing Coordinate in a Point given the Distance between It and Another Point

The coordinates of the points ๐ด, ๐ต, and ๐ถ are (๐พ, โˆ’2), (2, 8), and (โˆ’9, 6) respectively. Given that line ๐ด๐ต = line ๐ต๐ถ, find all possible values of ๐พ.

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Video Transcript

The coordinates of points ๐ด, ๐ต, and ๐ถ are ๐พ, negative two; two, eight; and negative nine, six, respectively. Given that line ๐ด๐ต is equal to line ๐ต๐ถ, find all possible values of ๐พ.

First, letโ€™s just sketch a coordinate plane to try and get a sense of whatโ€™s happening. Point ๐ด we donโ€™t know itโ€™s ๐‘ฅ-value; itโ€™s missing. But we do know that itโ€™s located at negative two for its ๐‘ฆ-value. Point ๐ด is located somewhere on this line. Point ๐ต is located at two, eight; point ๐ถ at negative nine, six. And this distance โ€” the distance from ๐ต to ๐ถ โ€” will be equal to the same distance from point ๐ด to point ๐ต.

Sketching this can be helpful, but it wonโ€™t give us an exact answer. To do that, weโ€™ll 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 ๐‘ฅ-values and ๐‘ฆ-values squared.

Weโ€™ll start out by finding the distance from point ๐ถ to point ๐ต. ๐‘ฅ two minus ๐‘ฅ one squared plus ๐‘ฆ two minus ๐‘ฆ 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 ๐ต and ๐ถ is the square root of 125. Weโ€™re 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 ๐ด to point ๐ต. Point ๐ด is at ๐พ, negative two. Point ๐ต is at two, eight. We can use point ๐ด as ๐‘ฅ one, ๐‘ฆ one and point ๐ต as ๐‘ฅ two, ๐‘ฆ two.

Plugging this into our equation, we get two minus ๐พ squared plus eight minus negative two squared. Two minus ๐พ squared cannot be simplified yet. But in place of eight minus negative two, we can say 10 squared. We notice that weโ€™re 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 ๐พ 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 ๐พ squared. To get rid of that squared, weโ€™ll take the square root of both sides of the equation.

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

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

Weโ€™ll do the same thing with the right equation, subtract two from both sides. Negative ๐พ is equal to negative seven. But we want positive ๐พ. So weโ€™ll multiply by negative one. And ๐พ is equal to seven.

At ๐พ equals negative three, the line ๐ด๐ต has a distance of the square root of 125. Thereโ€™s a possibility that ๐ด could be located at seven, negative two or negative three, negative two.

The possible values for ๐พ such that line ๐ด๐ต and line ๐ต๐ถ are equal is negative three or seven.

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