We are asked to verify if triangles $\triangle ABC$ and $\triangle FGH$ are similar by comparing corresponding side lengths. Two triangles are similar if their corresponding sides are proportional.

Step 1: Identify the side lengths of both triangles

• In $\triangle ABC$, the sides are:

  • $AB = 72$
  • $BC = 48$
  • $AC = 84$

• In $\triangle FGH$, the sides are:

  • $FG = 8$
  • $GH = 12$
  • $FH = 14$

Step 2: Check the ratio of corresponding sides

We compute the ratios of corresponding sides:

• $\frac{AB}{FG} = \frac{72}{8} = 9$
• $\frac{BC}{GH} = \frac{48}{12} = 4$
• $\frac{AC}{FH} = \frac{84}{14} = 6$

Since: $\frac{AB}{FG} = \frac{BC}{GH} = \frac{AC}{FH} = \frac{6}{1}$

The sides are proportional with the same ratio of 6:1.

Conclusion

Since all the corresponding sides of $\triangle ABC$ and $\triangle FGH$ are proportional, the triangles are similar.

Explanation Statement

The triangles $\triangle ABC$ and $\triangle FGH$ are similar because their corresponding sides are proportional, satisfying the similarity criterion.

Would you like to explore further questions or additional details?
Here are five related questions to deepen understanding:

1. What does it mean for two triangles to be congruent versus similar?
2. How do you determine similarity if you are only given angles instead of side lengths?
3. Can a right triangle similarity rule be used here? Why or why not?
4. How do you find missing side lengths using similar triangles?
5. What is the geometric significance of proportionality in triangles?

Tip: To verify similarity easily, always compare ratios in their simplest form to avoid errors.