Based on the figure, we can see that triangles OHJ and OIX are similar because they share the same angle at O and HI and HJ are tangent to the circle. Thus, we can set up the following proportion:
OH / OI = HJ / (HJ + 9)
Substituting the given values:
6.5 / 3.9 = HJ / (HJ + 9)
Cross-multiplying:
6.5(HJ + 9) = 3.9HJ
Distributing:
6.5HJ + 58.5 = 3.9HJ
Simplifying:
2.6HJ = -58.5
HJ = -22.5
However, this answer doesn't make sense in the context of the problem. Since HJ represents a length, it cannot be negative. Therefore, we need to re-examine our work.
Looking back at our proportion, we can see that the ratio of OH to OI is greater than 1, which implies that HJ is greater than 9. This makes sense since HJ is a chord of the circle and thus must be longer than the radius.
Let's try the problem again, but this time we'll rearrange the proportion to solve for HJ:
HJ / (HJ + 9) = OI / OH
Substituting the given values:
HJ / (HJ + 9) = 3.9 / 6.5
Cross-multiplying:
6.5HJ = 3.9HJ + 35.1
Simplifying:
2.6HJ = 35.1
HJ = 13.5
Therefore, HJ is equal to 13.5.