To solve for the corresponding side lengths in triangles \( \triangle ABC \) and \( \triangle XYZ \), we start by using the information that the triangles are similar. This means that the ratios of the lengths of corresponding sides are equal.
Given Side Lengths:
-
For triangle \( ABC \):
- \( AB = k \)
- \( BC = 9 \)
- \( CA = 4.5 \)
-
For triangle \( XYZ \):
- \( XY = 7 \)
- \( YZ = 6 \)
- \( ZX = 3 \)
Ratios of Corresponding Sides:
Since the triangles are similar, let's use two sides to write the ratios and solve for \( k \).
-
Comparing \( BC \) to \( YZ \): \[ \frac{BC}{YZ} = \frac{9}{6} = \frac{3}{2} \]
-
Comparing \( CA \) to \( ZX \): \[ \frac{CA}{ZX} = \frac{4.5}{3} = \frac{3}{2} \]
Both corresponding side pairs give us a ratio of \( \frac{3}{2} \).
Now we can set up a corresponding ratio for side \( AB \) to side \( XY \): \[ \frac{AB}{XY} = \frac{k}{7} = \frac{3}{2} \]
Solving for \( k \):
Cross-multiply to find \( k \): \[ 2k = 3 \times 7 \] \[ 2k = 21 \] \[ k = \frac{21}{2} = 10.5 \]
Thus, the value of \( k \) is \( 10.5 \).