Question

1. $\triangle abc sim \triangle def$. which statement(s) are correct? select all that apply. hint:make a sketch!

a) $\frac{bc}{ef}=\frac{ac}{df}$
b) $\frac{ab}{de}=\frac{ca}{fe}$
c) $\frac{ab}{ef}=\frac{bc}{de}$
d) $\frac{ca}{fd}=\frac{bc}{ef}$

Explanation:

Step1: Recall similar triangle ratios

For $\triangle ABC \sim \triangle DEF$, corresponding sides are proportional: $\frac{AB}{DE}=\frac{BC}{EF}=\frac{AC}{DF}$

Step2: Verify Option A

Check $\frac{BC}{EF}=\frac{AC}{DF}$. This matches the proportionality of corresponding sides (BC↔EF, AC↔DF).

Step3: Verify Option B

Check $\frac{AB}{DE}=\frac{CA}{FE}$. $CA=AC$, $FE=EF$; $\frac{AB}{DE}=\frac{AC}{EF}$ does not match the standard ratio (AC corresponds to DF, not EF).

Step4: Verify Option C

Check $\frac{AB}{EF}=\frac{BC}{DE}$. $AB$ corresponds to $DE$, $BC$ corresponds to $EF$, so this swaps non-corresponding sides and is incorrect.

Step5: Verify Option D

Check $\frac{CA}{FD}=\frac{BC}{EF}$. $CA=AC$, $FD=DF$, so $\frac{AC}{DF}=\frac{BC}{EF}$, which matches the proportionality of corresponding sides.

Answer:

A. $\frac{BC}{EF} = \frac{AC}{DF}$, D. $\frac{CA}{FD} = \frac{BC}{EF}$