The problem you provided asks to find the value of $HI$, given the following information:

• $FG \cong GH$ (meaning that segments $FG$ and $GH$ are congruent),
• $FI = w + 63$,
• $HI = 4w$.

Solution:

Since $FG \cong GH$, the triangles $\triangle FHI$ and $\triangle GHI$ are likely isosceles. Let's approach the problem with the information provided.

1. $HI = 4w$ is given.
2. $FI = w + 63$.

To find $HI$, we solve for $w$ using a relationship between the sides of the triangle. But from the given image and problem, we already know $HI = 4w$.

Thus, we already have $HI = 4w$, but the problem doesn't give additional numerical information to determine $w$ directly. If further information were provided, such as the value of $FI$ in terms of $FG$ or $GH$, we could proceed further.

However, without additional information, $HI = 4w$, and $w$ remains a variable in the equation.

Would you like more details on how to proceed or solve it with specific values of $w$?

Here are 5 questions related to this concept:

1. How do we use congruent triangles in geometry to solve for side lengths?
2. What properties define an isosceles triangle, and how can they help solve this problem?
3. Could a trigonometric approach be applied to calculate angles and side lengths here?
4. How can the Pythagorean theorem be used in situations involving perpendicular lines?
5. What role does symmetry play in solving problems with congruent line segments?

Tip: In geometry problems involving congruent sides, look for ways to use symmetry or congruent triangles to set up relationships between the variables.