Question

the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer. triangle x has ( mangle a = 30^circ ), ( mangle b = 90^circ ), and ( mangle c = 60^circ ) with side lengths of ( overline{ab} = 3 ), ( overline{bc} = 4 ), ( overline{ac} = 5 ). triangle y has ( mangle a = 30^circ ), ( mangle b = 90^circ ), and ( mangle c = 60^circ ) with side lengths of ( overline{ab} = 9 ), ( overline{bc} = 12 ), ( overline{ac} = 15 ). which statement best describes the relationship between these two triangles? ( \bigcirc ) they are similar triangles. ( \bigcirc ) they are congruent triangles. ( \bigcirc ) they are congruent but not similar triangles. ( \bigcirc ) they are similar and congruent triangles.

Explanation:

1. Recall the definitions:

• Similar triangles: Have the same shape (corresponding angles equal) and corresponding sides in proportion (same ratio).
• Congruent triangles: Have the same shape and size (corresponding angles equal and corresponding sides equal in length).

1. Analyze Triangle X and Triangle Y:

• Corresponding angles: Both have \( m\angle A = 30^\circ \), \( m\angle B = 90^\circ \), \( m\angle C = 60^\circ \). So corresponding angles are equal.
• Corresponding sides:
• For \( \overline{AB} \): Ratio is \( \frac{9}{3} = 3 \).
• For \( \overline{BC} \): Ratio is \( \frac{12}{4} = 3 \).
• For \( \overline{AC} \): Ratio is \( \frac{15}{5} = 3 \).
• All corresponding sides are in the same ratio (3:1), so sides are proportional.
• Check congruence: Corresponding sides are not equal (e.g., \( \overline{AB} \) is 3 vs. 9), so they are not congruent.
• Since angles are equal and sides are proportional, they are similar. They are not congruent because side lengths differ.

Answer:

They are similar triangles.