Question

If ΔABC
Δ
𝐴
𝐵
𝐶
is similar to ΔDEF
Δ
𝐷
𝐸
𝐹
, and BC=
𝐵
𝐶
=
36, EF=
𝐸
𝐹
=
3, and DE=
𝐷
𝐸
=
6, what is the measure of AB
𝐴
𝐵
?

Answer 1

To find the length of \( AB \) in triangle \( \Delta ABC \) given that it is similar to triangle \( \Delta DEF \), we can use the property of similar triangles which states that the ratios of corresponding sides are equal.

Given:
- \( BC = 36 \)
- \( EF = 3 \)
- \( DE = 6 \)

First, we need to find the scaling factor between the two triangles \( \Delta ABC \) and \( \Delta DEF \). We can use the sides \( DE \) and \( EF \):

The ratio between corresponding sides of similar triangles can be expressed as:

\[
\frac{DE}{EF} = \frac{6}{3} = 2
\]

This means that each side of triangle \( \Delta ABC \) is \( 2 \) times the corresponding side of triangle \( \Delta DEF \).

Now, we can set up a relationship for the sides \( BC \) and \( EF \):

\[
\frac{BC}{EF} = \frac{36}{3} = 12
\]

This suggests the ratio between triangle ABC and DEF sides is \( 12 : 1 \), indicating a scaling factor of \( 12 \).

Next, since we know that \( \frac{AB}{DE} \) must equal this scaling factor, we calculate \( AB \) by using the length of \( DE \):

Since \( DE = 6 \), and the scaling factor from smaller triangle \( \Delta DEF \) to larger triangle \( \Delta ABC \) is \( 12 \):

\[
AB = DE \cdot \text{scaling factor} = 6 \cdot 12 = 72
\]

Thus, the measure of \( AB \) is

\[
\boxed{72}
\]