Find the length of the side 𝐴𝐵.
To find the length of side 𝐴𝐵 in triangle 𝐴𝐵𝐶, we note first that this is a right triangle at 𝐴 and that 𝐷 is the perpendicular projection of 𝐴 onto side 𝐵𝐶. This being the case, we recall the right triangle altitude theorem, which we can use to find the length of side 𝐴𝐵. This gives us a formula for each of the squares of sides 𝐵𝐴 and 𝐶𝐴. And we see it’s the first of these that we can use to find side length 𝐴𝐵, since this is the same as 𝐵𝐴.
We have 𝐵𝐴 squared equals 𝐵𝐷 multiplied by 𝐵𝐶, where we see that 𝐵𝐶 is equal to the sum of 𝐵𝐷 and 𝐶𝐷. We’re given these lengths in the diagram. 𝐵𝐷 equals 9.6 and 𝐶𝐷 equals 5.4 centimeters. So, we have 𝐵𝐴 squared equal to 9.6 multiplied by the sum of 9.6 and 5.4. That’s 9.6 times 15, which is equal to 144. So, we have 𝐵𝐴 squared equal to 144. And taking the positive square root on both sides, positive since lengths are positive, we have 𝐵𝐴 equal to 12.
Since 𝐵𝐴 is the same as 𝐴𝐵, using the right triangle altitude theorem, we’ve been able to find the length of side 𝐴𝐵, which is equal to 12 centimeters.