### Given: Triangles ABC and DEF are SimilarIf \( DE = 6 \), \( DF = 9 \), \( AC = 27 \), and \( BC = 30 \), then what is the length of \( AB \)?The problem at hand involves two similar triangles, \( \triangle ABC \) and \( \triangle DEF \). Given the side lengths of triangle \( \triangle DEF \) and specific side lengths of triangle \( \triangle ABC \), we are to determine the length of the side \( AB \).#### Similar Triangles ExplanationWhen two triangles are similar, their corresponding angles are identical, and the lengths of their corresponding sides are proportional. Mathematically, this can be expressed with the following relationship:\[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} \]#### Given Measurements- \( DE = 6 \)- \( DF = 9 \)- \( AC = 27 \)- \( BC = 30 \)Using the proportionality relationship for similar triangles, we can set up the following equations to find the unknown side \( AB \):\[ \frac{AB}{DE} = \frac{AC}{DF} = \frac{BC}{EF} \]#### Solving for \( AB \)First, determine the scale factor between \( \triangle ABC \) and \( \triangle DEF \):\[ \frac{AC}{DF} = \frac{27}{9} = 3 \]This indicates that each side of \( \triangle ABC \) is three times the corresponding side of \( \triangle DEF \).Now, using the proportionality:\[ AB = DE \times 3 = 6 \times 3 = 18 \]#### Answer Choices- 12- 18- 81- 10Therefore, the correct answer is:\[ \boxed{18} \]### Graphical RepresentationThe provided diagram shows two triangles side by side labeled as \( \triangle ABC \) and \( \triangle DEF \). Triangle \( \triangle ABC \) is larger than triangle \( \triangle DEF \). Given the similarity between these triangles, their shapes are analogous, maintaining angle congruence and side proportionality.Feel free to explore more about similar triangles and their properties on our educational platform!

Name: ### Given: Triangles ABC and DEF are SimilarIf \( DE = 6 \), \( DF =
Uploaded: 2024-03-20T06:47:08.000Z
Channel: Bartleby
Description: ### Given: Triangles ABC and DEF are SimilarIf \( DE = 6 \), \( DF = 9 \), \( AC = 27 \), and \( BC = 30 \), then what is the length of \( AB \)?The problem at hand involves two similar triangles, \( \triangle ABC \) and \( \triangle DEF \). Given t