In right triangle abc below cd is the altitude to hypotenuse AB. If CD=6 and the ratio of AD to AB is 1:5 determine and state the length of BD

Question

Find the length of side BD.

Answer 1

When an altitude is drawn in a right triangle from the right angle to the hypotenuse, the square of the altitude is equal to the product of the two resultant parts of the hypotenuse. In this case the ratio of AD to AB is 1:5. This means the ratio of AD to DB must be 1:4. If we let x = AD, then 4x = DB. Let's multiply now:

 

(x)(4x) = 6^2

 

4x^2 = 36

 

x^2 = 9

 

x = 3

 

Therefore the length of BD is 4*3 = 12

Answer 2

Hi, Sally!

 

This problem uses the altitude-to-hypotenuse series of theorems. In this case, the altitude (CD) is the geometric mean between the two parts of the hypotenuse.

We don't know the two parts, but we know they are in a ratio of 1:5. This means that we can call one part x and the other 5x.

 

Using the geometric mean idea (which also comes from the two triangles being similar), we have

 

x / 6 = 6 / (5x).

 

Cross multiply to get

 

5x² = 36

x² = 36/5

x = 6 / √5 OR (6√5)/5 OR 2.6833

 

(we only need the positive, as a triangle can't have a side with a negative length).

 

So BD = 5 • 2.6833 or 13.4164.