Challenge: Distance between a point and a line
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Distance between point & line
Find the distance between the point negative 2, negative 4. This point right here. And the line y is equal to negative 1/3 x plus 2. That's this line right over here. Now to do it, we just need to figure out a perpendicular line to this blue line, to y is equal to negative 1/3 x plus 2, that contains this point right over here. So we need to figure out the equation of this line. And then we need to figure out where do these two lines intersect, and then we need to find the distance between these two points of intersection, and we have the shortest distance between this point and this line right over here. So the first step is to figure out what is the slope of this perpendicular line. Well, the slope of a perpendicular line is going to be the negative inverse of the slope of this blue line. So the negative inverse of negative 1/3 is going to be positive 3. So this line right over here is going to have a slope of 3. So it's going to have the form y is equal to 3x plus b, where b is its y-intercept. It looks, just eyeballing it here, that it's going to be pretty close to 2. But let's verify that. So to figure out what b actually is, let's substitute this point right over here. We know that not only is this line slope 3, but this point has to sit on it. So this point has to satisfy this equation. So when x is negative 2, y is negative 4. Or we have negative 4 is equal to 3 times negative 2 plus b. Let me write the negative 2 in there. 3 times negative 2 plus b. And now we can solve for b. We get negative 4 is equal to negative 6 plus b. Add 6 to both sides, you get 2 is equal to b or b is equal to 2. So we were right. The y-intercept for the second line is at 2. So we immediately can eyeball, or we can verify, where they both intersect. They both intersect the y-axis at y equals 2. For both of these, when x is equal to 0, y is equal to 2. If it wasn't so obvious, we could set these two equations equal to each other. We could say look, we have 3x plus 2. We know that this is now 3x plus 2, because b is 2. When does 3x plus 2 equal negative 1/3 x plus 2? Well, let's see. If we subtract 2 from both sides, when does 3x equal negative 1/3 x? Well, there's a couple of things that we could do right over here. We could add 1/3 x to both sides. And then we will get 3 and 1/3, which is the same thing as (10/3)x is equal to 0. And if you multiply both sides by 3/10, you get x is equal to 0. So these two lines intersect when x is equal to 0. For both of them, when x is equal to 0, y is equal to 2. But you could have eyeballed it here. You could have seen that both of their y-intercepts, which happens when x is equal to 0, y is equal to 2. So this point right over here is the point 0, 2. We already know that this point right over here is the point negative 2, negative 4. And now we just need to find the distance between these two points. And the distance formula really is just an application of the Pythagorean theorem. We just need to find the distance in the change in the y direction and the change in the x direction. So let's do that separately. So in the y direction, what is this distance right over here? So we went from y is equal to negative 4 to y is equal to 2. This distance right over here is 6. And what is this distance right over here? Well, we go from x equals negative 2 to x equals 0. So this distance right over here is 2. So the distance between these two points is really just the hypotenuse of a right triangle that has sides 6 and 2. If we call this distance d, we could say that the distance squared is equal to. And all I'm really doing here is restating the distance formula. The distance formula tells you all this Y2 minus Y1, which is 6, squared. But that's just the Pythagorean theorem. That's just saying 6 squared plus x2 minus x1, which is 0 minus negative 2, which is positive 2 squared is going to be equal to the distance squared. But we see that's just the Pythagorean theorem. But anyway, let's solve for the distance. So the distance squared is going to be equal to 36 plus 4, which is 40. And now, let's see. The distance is equal to the square root of 40. Square root of 40 is the same thing as the square root of 4 times 10. And so that's the same. So the distance is equal to 2 if we factor out the 4. It's the square root of 4 times square root of 10. 2 is the square root of 4. 2 square roots of 10. And we're done.