In the figure, the line segment 𝐴𝐵 lies in the plane 𝑥 and the line segment 𝐴𝐶 is perpendicular to 𝑥. Given that 𝐴𝐵 equals six and 𝐴𝐶 equals eight, find the length of the line segment 𝐵𝐶.
Let’s begin by adding what we know to our diagram. We’re told that 𝐴𝐵 is equal to six. So the line segment 𝐴𝐵 must be six units in length. Similarly, 𝐴𝐶 is equal to eight. So the line segment 𝐴𝐶 is eight units. We’re looking to find the length of the line segment 𝐵𝐶, as shown. So we need to use the final piece of information in our question. We’re told that the line segment 𝐴𝐵 lies in the plane 𝑥. And the line segment 𝐴𝐶 is perpendicular to that same plane. This means the line segment 𝐴𝐵 must be perpendicular to the line segment 𝐴𝐶. That is, 𝐴𝐵 and 𝐴𝐶 form a right angle.
I’ve redrawn the right-angle triangle so we can see what’s happening in a little bit more detail. Labelling the unknown side 𝑎, we see that we know the length of the short two sides in the right-angle triangle. And 𝑎 is the hypotenuse. We can therefore use the Pythagorean theorem to work out the length of the missing side. This says that the sum of the squares of the short two sides must be equal to the square of the longest side. It’s usually written as 𝑎 squared plus 𝑏 squared equals 𝑐 squared. Of course, in our triangle, the longest side is labelled 𝑎. So we could say that 𝑏 squared plus 𝑐 squared equals 𝑎 squared. Substituting the values from our triangle in, and we find that eight squared plus six squared equals 𝑎 squared.
Now, you might recognize a Pythagorean triple hit. But we’re going to solve the equation just in case. Eight squared is 64. And six squared is 36. Their sum is 100. So we find 𝑎 squared is equal to 100. We solve this equation by finding the square root of 100. Remember, we don’t need to find both a positive and negative square root since we’re dealing with a length. It absolutely must be positive. And when we find the square root of 100, we get 10. The length of 𝐵𝐶 is 10 units. Now, in fact, six, eight, and 10 form a Pythagorean triple. We know that six squared plus eight squared equals 10 squared. So we could’ve deduced quite quickly that 𝑎 was equal to 10. It is, of course, absolutely fine to solve the equation.