If `DeltaFGH` is isosceles and `FG lt 3cm, GH=8cm`, then of the following, the true relation is .

Options

A. `GH=FH`
B. `GF=GH`
C. `FH gt GH`
D. `GH lt GF`

Answer

To solve the problem, we need to analyze the properties of the isosceles triangle \( \Delta FGH \) given the conditions \( FG < 3 \, \text{cm} \) and \( GH = 8 \, \text{cm} \).

### Step-by-Step Solution:

1. **Understanding the Isosceles Triangle**:
In an isosceles triangle, two sides are equal. Let's denote the equal sides as \( FG \) and \( FH \). Therefore, we have:
\[ FG = FH \]

2. **Identifying the Given Values**:
From the problem, we know:
\[ FG < 3 \, \text{cm} \quad \text{and} \quad GH = 8 \, \text{cm} \]

3. **Setting Up the Equal Sides**:
Since \( FG = FH \), we can denote:
\[ FG = FH = x \quad \text{where} \quad x < 3 \, \text{cm} \]

4. **Applying the Triangle Inequality Theorem**:
The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Therefore, we can apply this to our triangle:
- For sides \( FG \) and \( GH \):
\[ FG + GH > FH \implies x + 8 > x \quad \text{(This is always true)} \]
- For sides \( FH \) and \( GH \):
\[ FH + GH > FG \implies x + 8 > x \quad \text{(This is also always true)} \]
- For sides \( FG \) and \( FH \):
\[ FG + FH > GH \implies x + x > 8 \implies 2x > 8 \implies x > 4 \, \text{cm} \]

5. **Contradiction**:
From the above inequality \( x > 4 \, \text{cm} \) contradicts our earlier condition \( x < 3 \, \text{cm} \). This means that the triangle cannot exist with the given conditions.

6. **Conclusion**:
Since \( FG < 3 \, \text{cm} \) and \( GH = 8 \, \text{cm} \), the only valid relation that can be concluded is:
\[ GH > FG \quad \text{and} \quad GH > FH \]
Therefore, the correct relation is:
\[ GH > FG \quad \text{and} \quad GH > FH \]