Use the image to answer the question.

Question

An illustration shows two similar triangles.
On triangle A B C, side A C measures 16, side A B measures 22, and side B C measures 24. Angle A measures 77 degrees, angle B measures 40 degrees, and angle C measures 63 degrees. On triangle D E F, side F D measures 8, side D E measures 11, and side E F measures 12. Angle D measures 77 degrees, angle E measures 40 degrees, and angle F measures 63 degrees.

ΔABC
  is similar to  ΔDEF
 . Find the ratio between  ΔDEF
  and  ΔABC
 . Express the answer as a fraction in simplest form.

(1 point)

GPT-4o mini · Answer

Triangles $ \Delta ABC $ and $ \Delta DEF $ are similar, which means that the corresponding sides are proportional. We can find the scale factor (ratio) between the two triangles by comparing the lengths of their corresponding sides. We will calculate the ratios for a pair of corresponding sides. From the information provided: - For side $ AC $ in $ \Delta ABC $ and side $ DF $ in $ \Delta DEF $: $$ \frac{DF}{AC} = \frac{8}{16} = \frac{1}{2} $$ - For side $ AB $ in $ \Delta ABC $ and side $ DE $ in $ \Delta DEF $: $$ \frac{DE}{AB} = \frac{11}{22} = \frac{1}{2} $$ - For side $ BC $ in $ \Delta ABC $ and side $ EF $ in $ \Delta DEF $: $$ \frac{EF}{BC} = \frac{12}{24} = \frac{1}{2} $$ All corresponding sides yield the same ratio of $ \frac{1}{2} $. Thus, the ratio of $ \Delta DEF $ to $ \Delta ABC $ is: $$ \frac{1}{2} $$ Expressed as a fraction in simplest form, the answer is: $$ \boxed{\frac{1}{2}} $$