Question

in circle O, BC =19 and DC=35. The diagram is not to scale.
what is the length of diameter BA?.

Answer 1

1. CD is tangent to the circle with center O, at point D. 

2. Join D to O. 

3. OD = OB = r, where r is the radius of the circle. 

4. radius OD is perpendicular to the tangent CD.

5. Thus in the right triangle OCD: $CD^{2} + DO^{2} = CO^{2}$, by the pythagorean theorem.

6. $(19+r)^{2}= 35^{2}+ r^{2}$

$38r=35 ^{2}-19^{2}$

Note: when applying pythagorean theorem, the difference of squares property
[ $a ^{2} -b^{2} =(a-b)(a+b)$ ] is a useful practical tool in calculations.

38r=(35 -19)(35+19)
38r=16*54

r= $\frac{16*54}{38}= 22.74$

7. Finally, the diameter BA=2r=2*22.74=45.48

So the answer is 45.5