The problem in the image involves intersecting chords in a circle. To find the length of $DH$, we can apply the chord-chord power theorem. This theorem states that if two chords intersect inside a circle, the products of the segments of each chord are equal. That is, for two intersecting chords $DE$ and $FG$, we have:

$DH \times HE = FH \times HG$

Given:

• $HE = 19$
• $FH = 15$
• $HG = 16$

We need to find $DH$.

According to the chord-chord theorem:

$DH \times 19 = 15 \times 16$

Now, let's solve for $DH$.The length of $DH$ is approximately $12.6$ when rounded to the nearest tenth.

Would you like further details or have any other questions?

Here are 5 related questions to explore:

1. What is the general proof of the chord-chord power theorem?
2. How would the problem change if the intersection point $H$ were outside the circle?
3. Can we solve for other segment lengths, such as $HE$, if only some information were given?
4. How does the chord-chord theorem relate to other circle theorems like the secant-tangent theorem?
5. What would happen if $H$ were the midpoint of both chords?

Tip: For problems involving circles, it's useful to remember multiple theorems such as the chord-chord, tangent-secant, and secant-secant power theorems.