From the figure, determine the length of the line 𝐵𝐷. If necessary, round your answer to the nearest hundredth.
We’re asked in this example to find the length of the line 𝐵𝐷, which we can see is the perpendicular projection to 𝐷 on the hypotenuse of the right triangle 𝐴𝐵𝐶 from the right angle at 𝐵. To find 𝐵𝐷, our first step will be to use the Pythagorean theorem to find the length of the hypotenuse 𝐴𝐶 of triangle 𝐴𝐵𝐶. We can then use this in one part of the right triangle altitude theorem to find the side length 𝐶𝐷 and then use this value in the Pythagorean theorem in right triangle 𝐶𝐷𝐵 to find the side length we want, which is 𝐵𝐷.
So, let’s start with the Pythagorean theorem applied to triangle 𝐴𝐵𝐶 to find the length of 𝐴𝐶. We have 𝐴𝐶 squared equal to 𝐶𝐵 squared plus 𝐴𝐵 squared. And substituting our two known side lengths, this gives 15 squared plus eight squared on the right-hand side. This evaluates to 289. And taking the positive square root on both sides, positive since lengths are positive, we have 𝐴𝐶 equals the square root of 289, which is 17 centimeters.
So, now marking this on the diagram and clearing some space, next we can use the right triangle altitude theorem to find side length 𝐶𝐷. We know that 𝐶𝐵 is 15 centimeters. And we’ve just found that 𝐴𝐶 is 17 centimeters. And so we have 15 squared equals 𝐶𝐷 multiplied by 17. Now, dividing through by 17 and evaluating 15 squared, we have 𝐶𝐷 equal to 225 over 17.
Next, applying the Pythagorean theorem to triangle 𝐶𝐷𝐵, we have 𝐵𝐷 squared equals 𝐵𝐶 squared minus 𝐶𝐷 squared. That’s 15 squared minus 225 over 17 all squared. This is 49.826 and so on. And taking the square root on both sides, we have 𝐵𝐷 equal to 7.058 and so on. Finally, rounding to the nearest hundredth, which is to two decimal places, we have that 𝐵𝐷 is equal to 7.06 centimeters.