In a right-angled triangle ABC, angle C is straight, CH is height. Find AH if AH: HB = 1: 8, AC = 9.

Given:
triangle ABC- rectangular
AC = 9
AH: HB = 1: 8 (can be written as a fraction)
Find: AH-?
Decision:
AH / HB = 1/8, this implies HB = 8 AH, AB = HB + AH = 9AN
Consider triangle ABC and triangle ASN
angle A – common
angle ANS = angle ABC = 90 degrees, since triangle ABC and triangle ASN are rectangular
therefore
triangle ABC is similar to triangle ACN
Since the ABC triangle is similar to the ACN triangle, AB / AC = AC / AH = 9AN / 9 = 9 / AH from this it follows that AH = 3
Answer: AH = 3.