The points F, G, H and I all lie on the same line segment, in that order, such that the ratio of F, G, colon, G, H, colon, H, IFG:GH:HI is equal to 4, colon, 4, colon, 3, .4:4:3. If F, I, equals, 33, commaFI=33, find F, H, .FH.


We are given that points \( F \), \( G \), \( H \), and \( I \) lie on a straight line segment, and the ratio of the distances between these points is \( FG : GH : HI = 4:4:3 \). Additionally, we know that the total length \( FI = 33 \).

### Step 1: Set up the ratio

The ratio \( FG : GH : HI = 4:4:3 \) tells us the relative lengths of the segments \( FG \), \( GH \), and \( HI \). If we assign a common factor \( x \) to these segments, we can express their lengths as:

\[
FG = 4x, \quad GH = 4x, \quad HI = 3x
\]

### Step 2: Calculate the total length

The total length from \( F \) to \( I \) is given as \( FI = 33 \). From the diagram, \( FI \) consists of the sum of the three segments \( FG \), \( GH \), and \( HI \), so:

\[
FI = FG + GH + HI = 4x + 4x + 3x = 11x
\]

Since \( FI = 33 \), we can now solve for \( x \):

\[
11x = 33
\]
\[
x = \frac{33}{11} = 3
\]

### Step 3: Find the length of \( FH \)

The distance \( FH \) is the sum of \( FG \) and \( GH \):

\[
FH = FG + GH = 4x + 4x = 8x
\]

Substitute \( x = 3 \) into this equation:

\[
FH = 8 \times 3 = 24
\]

Thus, the length of \( FH \) is \( \boxed{24} \).

### Would you like more details on this solution?

#### 5 Related Questions:
1. How can you generalize this method to solve problems with different ratios?
2. What happens if the total length \( FI \) changes but the ratios remain the same?
3. How can we verify that the points \( F, G, H, I \) are correctly placed on the segment?
4. What if an additional point \( J \) was added in the same ratio pattern? How would we find \( FJ \)?
5. How does this problem change if the points form a non-linear configuration?

#### Tip:
Always start by converting ratios into algebraic expressions to simplify the problem and find unknown values easily.

The points F, G, H and I all lie on the same line segment, in that order, such that the ratio of F:G:H:I is equal to 4:4:3. If FI = 33, find FH.

Solve for the length of segment FH on a line segment where the ratio FG:GH:HI is 4:4:3 and FI = 33. This problem involves key algebraic concepts such as ratios and linear segments. Suitable for students in Grades 6-8.

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