If ΔABC is similar to ΔDEF , and AB= 16, DF= 3, and AC= 12, what is the length of DE ?

1 answer

Since triangles ΔABC and ΔDEF are similar, the corresponding sides are in proportion. This means that the ratio of the lengths of corresponding sides is constant.

Given:

  • \( AB = 16 \)
  • \( DF = 3 \)
  • \( AC = 12 \)
  • Let \( DE \) be the length we need to find.

The ratio of the sides can be represented as:

\[ \frac{AB}{DF} = \frac{AC}{DE} \]

Substituting the known values into the equation, we have:

\[ \frac{16}{3} = \frac{12}{DE} \]

Now, we can cross-multiply to solve for \( DE \):

\[ 16 \cdot DE = 3 \cdot 12 \]

Calculating the right side:

\[ 16 \cdot DE = 36 \]

Now, divide both sides by 16 to isolate \( DE \):

\[ DE = \frac{36}{16} = \frac{9}{4} = 2.25 \]

Thus, the length of \( DE \) is \( \frac{9}{4} \) or \( 2.25 \).