Question Video: Finding the Distance between a Point and the 𝑥-Axis given the Point’s Coordinates Mathematics • Third Year of Secondary School

Video Transcript

What is the distance between the point 19, five, five and the 𝑥-axis?

Any point that lies on the 𝑥-axis will have coordinates 𝑥, zero, zero. Both the 𝑦- and 𝑧-coordinates must be equal to zero. We’re given the coordinates of a point 19, five, five. The point on the 𝑥-axis that is closest to this will have coordinates 19, zero, zero. The shortest distance will be to the point where the 𝑥-coordinate is the same. We know that we can calculate the distance between two points in three dimensions using an adaption of the Pythagorean theorem. If we have two points with coordinates 𝑥 one, 𝑦 one, 𝑧 one and 𝑥 two, 𝑦 two, 𝑧 two, the distance between them is equal to the square root of 𝑥 two minus 𝑥 one squared plus 𝑦 two minus 𝑦 one squared plus 𝑧 two minus 𝑧 one squared.

Substituting in our two coordinates gives us the square root of 19 minus 19 squared plus zero minus five squared plus zero minus five squared. 19 minus 19 is equal to zero. Zero minus five is equal to negative five. So we are left with the square root of negative five squared plus negative five squared. Multiplying a negative number by a negative number gives us a positive answer. Therefore, negative five squared is equal to 25. This means that our answer simplifies to the square root of 50.

It is worth pointing out that we could have subtracted the coordinates in the other order as five minus zero squared is also equal to 25. As squaring a number always gives a positive answer, it doesn’t matter which order we subtract our coordinates in. We can actually simplify our answer by using our laws of radicals or surds. The square root of 50 is equal to the square root of 25 multiplied by the square root of two. As the square root of 25 equals five, we’re left with five multiplied by the square root of two or five root two. The square root of 50 is equal to five root two. We can therefore conclude that the distance between the points 19, five, five and the 𝑥-axis is five root two length units.

We might actually notice a shortcut here. To find the distance between any point and an axis, we simply find the sum of the squares of the other two coordinates and then square root the answer. As we want to calculate the distance to the 𝑥-axis, we square the 𝑦- and 𝑧-coordinates, find their sum, and then square root our answer. If we needed to calculate the distance between a point and the 𝑦-axis, we would square the 𝑥- and 𝑧-coordinates, find the sum of these, and then square root that answer. We would use the same method to find the distance between a point and the 𝑧-axis, this time using the 𝑥- and 𝑦-coordinates.