Question

which two triangles are similar? abc & def none are similar

Explanation:

Step1: List triangle side lengths

Triangle ABC: $AB=6$, $BC=9$, $AC=12$
Triangle DEF: $DE=4$, $EF=6$, $FD=8$
Triangle GHJ: $GH=6$, $HJ=10$, $GJ=14$

Step2: Simplify side ratios for ABC

$\frac{AB}{BC}=\frac{6}{9}=\frac{2}{3}$, $\frac{BC}{AC}=\frac{9}{12}=\frac{3}{4}$, $\frac{AB}{AC}=\frac{6}{12}=\frac{1}{2}$

Step3: Simplify side ratios for DEF

$\frac{DE}{EF}=\frac{4}{6}=\frac{2}{3}$, $\frac{EF}{FD}=\frac{6}{8}=\frac{3}{4}$, $\frac{DE}{FD}=\frac{4}{8}=\frac{1}{2}$

Step4: Simplify side ratios for GHJ

$\frac{GH}{HJ}=\frac{6}{10}=\frac{3}{5}$, $\frac{HJ}{GJ}=\frac{10}{14}=\frac{5}{7}$, $\frac{GH}{GJ}=\frac{6}{14}=\frac{3}{7}$

Step5: Match equal ratio sets

ABC and DEF have identical simplified side ratios, so they are similar by SSS similarity.

Answer:

ABC & DEF