The coordinates of points 𝐴, 𝐵, and 𝐶 are 𝐾, negative two; two, eight; and
negative nine, six, respectively. Given that line 𝐴𝐵 is equal to line 𝐵𝐶, find all possible values of 𝐾.
First, let’s just sketch a coordinate plane to try and get a sense of what’s
happening. Point 𝐴 we don’t know it’s 𝑥-value; it’s missing. But we do know that it’s located at negative two for its 𝑦-value. Point 𝐴 is located somewhere on this line. Point 𝐵 is located at two, eight; point 𝐶 at negative nine, six. And this distance — the distance from 𝐵 to 𝐶 — will be equal to the same distance
from point 𝐴 to point 𝐵.
Sketching this can be helpful, but it won’t give us an exact answer. To do that, we’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 𝑥-values and 𝑦-values squared.
We’ll start out by finding the distance from point 𝐶 to point 𝐵. 𝑥 two minus 𝑥 one squared plus 𝑦 two minus 𝑦 one squared. Two minus negative nine equals 11 — we keep our square, 11 squared — 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 𝐵 and 𝐶 is the square root of 125. We’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 𝐴 to point 𝐵. Point 𝐴 is at 𝐾, negative two. Point 𝐵 is at two, eight. We can use point 𝐴 as 𝑥 one, 𝑦 one and point 𝐵 as 𝑥 two, 𝑦 two.
Plugging this into our equation, we get two minus 𝐾 squared plus eight minus
negative two squared. Two minus 𝐾 squared cannot be simplified yet. But in place of eight minus negative two, we can say 10 squared. We notice that we’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 𝐾 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 𝐾 squared. To get rid of that squared, we’ll take the square root of both sides of the
Here’s where we really have to pay attention. The square root of 25 is five. But it’s also negative five. We need plus or minus five is equal to two minus 𝐾. We need to break this equation up into two pieces: two minus 𝐾 equals positive five
and two minus 𝐾 equals negative five.
For the left equation, I’ll subtract two from both sides. This leaves me with negative 𝐾 equals three. But I’m not interested in negative 𝐾. I want to know what positive 𝐾 is. And that means I’ll multiply the whole equation by negative one. 𝐾 is equal to negative three.
We’ll do the same thing with the right equation, subtract two from both sides. Negative 𝐾 is equal to negative seven. But we want positive 𝐾. So we’ll multiply by negative one. And 𝐾 is equal to seven.
At 𝐾 equals negative three, the line 𝐴𝐵 has a distance of the square root of
125. There’s a possibility that 𝐴 could be located at seven, negative two or negative
three, negative two.
The possible values for 𝐾 such that line 𝐴𝐵 and line 𝐵𝐶 are equal is negative
three or seven.