Answer:
25cm
Step-by-step explanation:
**see attached diagram**
Let BH = h (height)
OH = radius r = (5/17)h ⇒ h = (17/5)r
Area ∆ABC = 1/2 × base × height = 1/2 x AC x h
substituting h = (17/5)r: = 1/2 x AC x (17/5)r
= 17/10 × AC × r
We can split the isosceles triangle into three separate triangles indicated by the red lines on the diagram attached. Because the radius always meets a tangent (points E, D and H) at a right angle, the area of each triangle will be the length of the side multiplied by the radius of the circle:
Area ∆BOC = Area ∆BOA = 1/2 × 30 × r = 15r
Area ∆AOC = 1/2 × AC × r = 1/2 × AC × r
Therefore, area ∆ABC = ∆AOC + ∆BOC + ∆BOA
=1/2 (AC)r + 15r + 15r
=1/2 (AC)r + 30r
Now we have 2 different equations for the area of the isosceles triangle ABC. Equate both equations and solve to find AC:
Area ∆ABC = Area ∆ABC
17/10 (AC)r = 1/2 (AC)r + 30r
Divide both sides by r: 17/10 (AC) = 1/2 (AC) + 30
Collect & combine like terms: 6/5 (AC) = 30
Divide by 6/5: AC = 25 cm
