Question Video: Finding the Distance between Two Given Points by Using the Distance between Two Points Formula Mathematics • 8th Grade

Video Transcript

Find the distance between points 𝐴 and 𝐵.

We could solve this a few ways. One of the ways would be using the distance formula. For two points 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two, the distance between them can be found by taking the square root of 𝑥 two minus 𝑥 one squared plus 𝑦 two minus 𝑦 one squared. So we can let 𝐴 be 𝑥 one, 𝑦 one and 𝐵 be 𝑥 two, 𝑦 two. So these are points. And we can find them here on our grid. 𝐴 is located at negative three for 𝑥 and four for 𝑦. So 𝐴 is the point negative three, four. And 𝐵 is at zero for 𝑥 and negative three for 𝑦. So 𝐵 is the point zero, negative three.

Let’s go ahead and plug in 𝐴 first, the 𝑥 one, 𝑦 one. So we need to plug in negative three for 𝑥 one. And we need to plug in four for 𝑦 one. Now let’s do the same for 𝐵. 𝑥 two is zero. And 𝑦 two is negative three. So we replace 𝑥 two with zero and 𝑦 two with negative three. And now we can solve. So when solving, we need to work on the innermost parenthesis, which there are two. So zero minus negative three, two negatives make a positive, so it’s really zero plus three. So we have three squared. And then we have negative three minus four. That’s negative seven. And now we need to square these numbers. Three squared is nine. And negative seven squared is 49. When we square a number, whether it’s positive or negative, it will be turned positive when squaring it. Nine plus 49 is 58. Therefore, the distance between points 𝐴 and 𝐵 will be square root of 58 length units. Therefore, our final answer will be square root of 58 length units.

Now we also could solve this problem using triangles. If we could create a right triangle using 𝐴 and 𝐵, we could use the Pythagorean theorem to find the missing length, the distance between them. So we could find this distance between 𝐴 and 𝐵, calling it 𝑥, if it were a side length of a right triangle. So this could be a side. And this could be a side. And we know those lengths using our grid. This short length would be three. And the longer length would be four plus three. So it would be seven. And the right angle would be found here because the 𝑥- and 𝑦-axis are perpendicular.

So here’s our triangle. The Pythagorean theorem states: the square of the longest side is equal to the sum of the squares of the shorter sides. The longest side is always across from the 90-degree angle. It’s called the hypotenuse. So 𝑥 will be our longest side. And the other two will be our shorter sides. So let’s go ahead and plug these in. So 𝑥 squared is equal to three squared plus seven squared. Three squared is nine. And seven squared is 49. Now adding nine and 49, we get — and we have that 𝑥 squared is equal to 58 when adding nine and 49. So now, we square root both sides. And we get 𝑥 is equal to the square root of 58, just like before.

So the distance between points 𝐴 and 𝐵 will be square root 58 length units.