To solve for the missing side \( x \) in the triangle \( \Delta PAN \) given that it is similar to another triangle \( \Delta \DOT P \), we can use the properties of similar triangles. The ratios of corresponding sides of similar triangles are equal.
Given the sides of triangle \( \Delta PAN \):
- Side \( PA = 21 , m \)
- Side \( AN = 14 , m \)
- Side \( PN = 54 , m \)
- Side \( DT = x , m \)
Since the triangles are similar, we can set up the following proportion using corresponding sides:
\[ \frac{PA}{DT} = \frac{AN}{TO} = \frac{PN}{OT} \]
If we let \( TO \) represent the side corresponding to \( x \), we will use just two sides for this calculation.
Set up the proportion using \( PA \) and \( DT \):
\[ \frac{21}{x} = \frac{14}{54} \]
Now we can solve for \( x \). First, cross multiply:
\[ 21 \times 54 = 14 \times x \]
Calculating \( 21 \times 54 \):
\[ 21 \times 54 = 1134 \]
Now the equation is:
\[ 1134 = 14x \]
Next, divide both sides by 14 to solve for \( x \):
\[ x = \frac{1134}{14} = 81 \]
Thus, the value of \( x \) is:
\[ x = 81 , m \]
So the missing side \( x \) is \( 81 , m \).