Video Transcript
Which of the following points is at a distance of five root two from the origin? (A) Zero, five; (B) five, zero; (C) five root two, five root two; or (D) five, five.
So we’ve been given the coordinates of four points and asked to determine which of these points is a given distance of five root two units from the origin. Let’s consider how we would find the distance of a general point with coordinates 𝑥, 𝑦 from the origin. We’ll label this straight-line distance as 𝑑. We can sketch in a right triangle, and the distance of the point from the origin is then the length of the hypotenuse of this triangle. If we know the coordinates of this point, we can calculate the value of 𝑑 by applying the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.
Now, the length of the horizontal side in this particular triangle is 𝑥 because that’s the distance along the 𝑥-axis. And the length of the vertical side is 𝑦; that’s the distance along the 𝑦-axis. So by applying the Pythagorean theorem, we have 𝑑 squared is equal to 𝑥 squared plus 𝑦 squared. Taking the square root of both sides of this equation, we then have that 𝑑 is equal to the square root of 𝑥 squared plus 𝑦 squared.
Now, suppose instead that the point were in a different quadrant. Suppose, for example, that it was in the second quadrant, where 𝑥 is negative and 𝑦 is positive. This time, the length of the horizontal side in the triangle would be the absolute value of 𝑥. But when we square the absolute value of 𝑥, this is the same as 𝑥 squared. So the formula that we’ve just written down will work no matter which quadrant the point we’re interested in is in or even if it’s on either of the coordinate axes.
In fact, this formula is a specific case of a more general formula for calculating the distance between any two points. Suppose we have two points with coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two. If we draw in a right triangle below the line connecting these two points, then the length of the horizontal side will be the difference between the 𝑥-coordinates. That’s 𝑥 two minus 𝑥 one. And the length of the vertical side will be the difference between the 𝑦-coordinates. That’s 𝑦 two minus 𝑦 one. Then, applying the Pythagorean theorem, we have 𝑑 squared is equal to 𝑥 two minus 𝑥 one squared plus 𝑦 two minus 𝑦 one squared. For this reason, it doesn’t matter which way around we subtract the 𝑥- and 𝑦-coordinates, because whichever way we subtract, when we square, we’ll get the same result.
Square rooting both sides, we have that 𝑑 is equal to the square root of 𝑥 two minus 𝑥 one squared plus 𝑦 two minus 𝑦 one squared. And this is the general distance formula for finding the distance between any two points on a coordinate grid, which we should commit to memory. Now, in this problem, we’re specifically finding the distance between each point and the origin, which has the coordinates zero, zero. So substituting zero for 𝑥 one and 𝑦 one and 𝑥, 𝑦 for the coordinates of each point, we can see that the distance formula will simplify to the formula we wrote down earlier. So all we need to do is substitute the coordinates of each point into this formula and see which gives us an answer of five root two.
Let’s start with option (A) then. And we have that the distance between this point and the origin, 𝑑 sub 𝐴, is equal to the square root of zero squared plus five squared. That’s the square root of zero plus 25 or simply the square root of 25, which is five. So point (A) is not the correct distance from the origin.
Let’s consider point (B). For point (B), the distance is the square root of five squared plus zero squared. Once again, that’s the square root of 25, which is equal to five. In fact, for each of these points, we can see that their distance from the origin is five units by simply sketching them on a coordinate grid, as each point lies on one of the coordinate axes.
Next, let’s consider point (C). And this time, the distance from the origin is equal to the square root of five root two squared plus five root two squared. Remember, five root two squared means five root two multiplied by five root two, which we can write as five times five times root two times root two. That’s 25 times two, which is equal to 50, so distance (C) is the square root of 50 plus 50. That’s the square root of 100, which is equal to 10. None of the points so far are the correct distance from the origin. So let’s check the final point.
For point (D), which has coordinates five, five, the distance from the origin is the square root of five squared plus five squared. That’s the square root of 25 plus 25, which is the square root of 50. Now, we need to consider how we can simplify this surd or radical. To simplify radicals, we look for square factors, and of course 50 is equal to 25 multiplied by two. So the square root of 50 is the square root of 25 times two. We can then write this as the product of the radicals. It’s equal to the square root of 25 multiplied by the square root of two. And as the square root of 25 is five, this simplifies to five root two.
So point (D) is the point we’re looking for. It’s the point that is five root two units from the origin. Using a special case of the distance formula then, we found that the point that is a distance of five root two units from the origin is the point five, five.
