To find the value of \( DP \) in triangle \( DRP \), we can use the Triangle Inequality Theorem. The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
Here, we have:
- \( RP = 5.8 \)
- \( DR = 9.6 \)
- We need to find \( DP \).
According to the Triangle Inequality Theorem:
- \( DR + RP > DP \)
- \( DR + DP > RP \)
- \( RP + DP > DR \)
Let's substitute the known values into these inequalities.
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From \( DR + RP > DP \): \[ 9.6 + 5.8 > DP \quad \Rightarrow \quad 15.4 > DP \]
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From \( DR + DP > RP \): \[ 9.6 + DP > 5.8 \quad \Rightarrow \quad DP > 5.8 - 9.6 \quad \Rightarrow \quad DP > -3.8 \quad (\text{which is always true}) \]
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From \( RP + DP > DR \): \[ 5.8 + DP > 9.6 \quad \Rightarrow \quad DP > 9.6 - 5.8 \quad \Rightarrow \quad DP > 3.8 \]
Combining the results from the inequalities, we find that: \[ 3.8 < DP < 15.4 \]
The options given are \( 7.6, 3.8, 15.4, 11.2 \). Of these options, only \( 7.6 \), \( 11.2 \), and \( 15.4 \) fit our \( DP > 3.8 \) condition and \( DP < 15.4 \).
Additionally, \( 15.4 \) is not a possibility because it does not satisfy \( DP < 15.4 \).
Thus the valid options for \( DP \) are \( 7.6 \) and \( 11.2 \).
Since we need to determine a single value:
If we select \( 7.6 \), it fits all triangle inequalities as:
From the second inequality \( DP = 7.6 \):
- \( DR + DP = 9.6 + 7.6 = 17.2 > RP = 5.8 \)
- \( DR + RP = 9.6 + 5.8 = 15.4 > DP = 7.6 \)
- \( RP + DP = 5.8 + 7.6 = 13.4 > DR = 9.6 \)
Therefore, all conditions are satisfied.
Thus the value of \( DP \) is \( \boxed{7.6} \).