Find the length of 𝐴𝐹 approximating the result to the nearest hundredth.
In this example, we’re asked to find the length of 𝐴𝐹, where in right triangle 𝐴𝐵𝐷 𝐴𝐹 is the perpendicular projection from the right angle at 𝐴 to 𝐹 on side 𝐵𝐷.
The first thing we can do to get started is use the Pythagorean theorem to find the length of side 𝐷𝐵 in right triangle 𝐴𝐵𝐷. In our case, this gives 𝐴𝐵 squared plus 𝐴𝐷 squared is equal to 𝐷𝐵 squared. That is, 33 squared plus 56 squared equals 𝐷𝐵 squared. Evaluating the left-hand side gives 𝐷𝐵 squared equal to 4225. And taking the positive square root on both sides, we have 𝐷𝐵 equal to 65.
Okay, so we have the side length 𝐷𝐵, but how does this help us to find 𝐴𝐹? Well, our next step is to use the right triangle altitude theorem to find the two side lengths 𝐷𝐹 and 𝐵𝐹. We can do this since we have lengths 𝐴𝐷, 𝐴𝐵, and 𝐷𝐵, which is the same as 𝐵𝐷. And that’s all of the other lengths involved in these two equations. And the reason we want to do this is so we can use 𝐷𝐹 and 𝐵𝐹 in the corollary of the altitude theorem to find the length we want, which is 𝐴𝐹.
So let’s start with the first equation. That’s 𝐴𝐷 squared equals 𝐷𝐹 times 𝐷𝐵. In our case, this gives us 56 squared equals 𝐷𝐹 times 65. Then, evaluating the square on the left-hand side and dividing both sides by 65, we find 𝐷𝐹 equal to 48.2461 and so on. Next, looking at the second equation in the altitude theorem, 𝐴𝐵 squared equals 𝐵𝐹 multiplied by 𝐵𝐷, we have 33 squared equal to 𝐵𝐹 times 65. Again, evaluating the square on the left and dividing through by 65, we have that 𝐵𝐹 is equal to 16.7538 and so on.
So now we have 𝐷𝐹 and 𝐵𝐹, which are the two values we need to find the length 𝐴𝐹 using the corollary. Substituting these values in, we have 𝐴𝐹 squared equal to 48.2461 multiplied by 16.7538. This is 808.3086 and so on. And now taking the positive square root on both sides, we have 𝐴𝐹 equal to 28.4307.
And so since we’re asked to approximate the length of 𝐴𝐹 to the nearest hundredth, which is to two decimal places, using the Pythagorean theorem together with the right triangle altitude theorem and its corollary, we find the length 𝐴𝐹 to the nearest hundredth equals 28.43 centimeters.