The distance between the points (2,1) and (x,7) as graphed on the standard (x,y) coordinate plane is 10. What is one possible value for x?
Pro tip: when you see questions about the distance between points, remember that the distance formula (We all love the distance formula, right? Blech.) is based on the Pythagorean Theorem! If the distance between two points that don’t have the same or coordinate is 10, a nice even number, then the points must make a nice Pythagorean Triple. Because the distance here is 10, I’m thinking it’ll be a 6-8-10. Let’s see.
Here we have our given point, (2, 1), the = 7 line (because we don’t know what the -value is in the other point, and the vertical distance between the two, which happens to be 6.
All we need to do here is figure out where on the dotted line a point can go that will be exactly 10 units away from (2, 1), and because we know that 6-8-10 is a Pythagorean triple, and we already have a distance of 6, we just need to move either 8 spaces to the right or 8 spaces to the left of (2, 7) on the line. Or even better, because this is a multiple choice question, we just need to evaluate which answer choice is 8 units away from 2.
Of course, it’s -6. (-6, 7) is a horizontal distance of 8 away from (2, 7), and boom, we’ve got our 6-8-10 triangle.