Question

Find the length of the altitude to the hypotenuse under the given conditions. Give the answer in simplest radical form.

Answer 1

Answer:

AD = 2√10

Step-by-step explanation:

We're looking for the length AD.

To start, let's break off two smaller triangles from this large one: ΔBDA andΔADC. These triangles are similar: that is, all their interior angles are the equal. Since they're similar, the corresponding sides of these triangles will be in proportion to each other. That is to say, the ratio between corresponding sides will be equal. AD on the larger triangle corresponds to BD on the smaller one, and DC on the larger triangle corresponds to DA on the smaller one. Using this information, we can set up the ratio

$\dfrac{AD}{BD} =\dfrac{DC}{DA}$

AD and DA describe the same line segment, so we can replace DA with AD, and substituting BD = 5 and DC = 8 makes the equation

$\dfrac{AD}{5} =\dfrac{8}{AD}$

Cross multiplying gives us $(AD)^2=40$, and taking the square root gives us our value for AD:

$\sqrt{(AD)^2}=\sqrt{40}\\AD=\sqrt{4\cdot10}=\sqrt{4}\sqrt{10}=2\sqrt{10}$

So our solution is AD = 2√10