If `DeltaFGH `is isosceles and `FG lt 3 cm, GH = 8 cm`, 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 given information about triangle FGH, which is isosceles with the sides FG and GH specified.

### Step-by-Step Solution:

1. **Identify the Sides of the Triangle**:
- We know that triangle FGH is isosceles. This means that at least two sides are equal.
- We are given that FG < 3 cm and GH = 8 cm.

2. **Understanding Isosceles Triangle Properties**:
- In an isosceles triangle, the two equal sides can either be FG and FH or FG and GH or GH and FH.
- Since FG is less than 3 cm and GH is 8 cm, it is not possible for FG to be equal to GH because FG is shorter.

3. **Determine the Possible Equal Sides**:
- Since FG cannot be equal to GH (as FG < 3 cm and GH = 8 cm), the only possibility left is that FG = FH.
- Therefore, we can conclude that FG = FH.

4. **Using 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.
- For triangle FGH, we can write the inequalities:
- FG + GH > FH
- FG + FH > GH
- GH + FH > FG

5. **Substituting Known Values**:
- Since FG = FH, we can denote FG as x (where x < 3 cm).
- So, we have:
- x + 8 > x (which is always true)
- x + x > 8 → 2x > 8 → x > 4 (not possible since x < 3)
- 8 + x > x (which is always true)

6. **Conclusion**:
- The only valid relationship we can derive from the given conditions is that FG = FH.
- Therefore, the true relation is FG = FH.

### Final Answer:
The true relation is FG = FH.