In the right △ABC, CH is the altitude to the hypotenuse (H∈ AB). Find the ratio of AH:HB, if m∠BAC=30°.

In a 30°-60°-90° triangle, the hypotenuse is twice the length of the short leg. Here, it is convenient to assign segment BH the length of 1.

BH is the short leg of ∆BCH, so BC will have length 2.

BC is the short leg of ∆BAC, so BA will have length 4.

Now, we have

... (AH + HB) : HB = 4 : 1

So the ratio we want is ...

... AH : HB = 3 : 1