In the figure at the right, \( D F=22, B C=6 \), and \( D B=8 \). Find the perimeter of \( \triangle A D F \)

To find the perimeter of triangle \( ADF \), we need to find the lengths of all three sides: \( AD \), \( DF \), and \( AF \). We know that \( DF = 22 \). Now, to find \( AD \) and \( AF \), we can use the information given about points \( B \) and \( C \). Given \( DB = 8 \) and \( BC = 6 \), we can find \( DC \) since \( D \) is point \( D - B \). This means: \[ DB + BC = DC \implies 8 + 6 = 14 \implies DC = 14 \] Next, we need to connect this information to find side lengths associated with triangle \( ADF \). Assuming that points \( A \), \( D \), and \( F \) line up in such a way that we can use the lengths we've established, we can focus on calculating \( AD \) and \( AF \), likely based on spatial arrangement or additional provided specifics. However, with the information provided, we still only have \( DF \) at \( 22 \). In a typical right triangle or where \( AD \) and \( AF \) relate to \( DB \) or \( DC \), additional geometric specifics would assist us in establishing concrete lengths. Given our known length of \( DF = 22 \), one could assume typical relationships if \( AF = AD \) from symmetry or equal length aspects yet uncalculated with the given information. Thus, if we can assume some relationships among lengths (akin to similar triangles or proportions), we can enlist \( AD + AF = 8+ DC\) depending upon your spatial frame. To find that perimeter: \[ Perimeter = AD + DF + AF \] If we had \( AD \) or \( AF\) at say proportions of existing known lengths, we could easily substitute and calculate. For now, with only \( DF \) as our quantifiable, sorry, we are left hanging on the specifics there. What’s the missing puzzle piece? In common cases or streets, exploration usually elucidates other lengths.