Triangle ABC has two sides that measure 12 inches and 16 inches. Which could be the measures of the corresponding sides of a triangle that is similar to triangle ABC?
1. 10 in and 14 in
2. 9 in and 12 in
3. 6 in and 10 in
4. 3 in and 8 in
To determine which could be the measures of the corresponding sides of a triangle similar to triangle ABC, we need to understand the concept of similar triangles.
Two triangles are similar if their corresponding angles are congruent and the ratios of their corresponding side lengths are equal.
In triangle ABC, we have two sides with lengths of 12 inches and 16 inches. Let's assume that the two sides with lengths 12 inches and 16 inches are the corresponding sides of the similar triangle we are looking for.
Now, we compare the corresponding side lengths of the original triangle with the options given:
1. 10 in and 14 in: The ratio of corresponding sides in the original triangle is 12/16 = 3/4. However, the ratio of the corresponding sides in the first option is 10/14 = 5/7, which is not equal to the ratio in the original triangle. Therefore, it is not possible for this option to be the measures of the corresponding sides of a similar triangle.
2. 9 in and 12 in: The ratio of corresponding sides in the original triangle is 12/16 = 3/4. However, the ratio of the corresponding sides in the second option is 9/12 = 3/4, which is equal to the ratio in the original triangle. Therefore, it is possible for this option to be the measures of the corresponding sides of a similar triangle.
3. 6 in and 10 in: The ratio of corresponding sides in the original triangle is 12/16 = 3/4. However, the ratio of the corresponding sides in the third option is 6/10 = 3/5, which is not equal to the ratio in the original triangle. Therefore, it is not possible for this option to be the measures of the corresponding sides of a similar triangle.
4. 3 in and 8 in: The ratio of corresponding sides in the original triangle is 12/16 = 3/4. However, the ratio of the corresponding sides in the fourth option is 3/8, which is not equal to the ratio in the original triangle. Therefore, it is not possible for this option to be the measures of the corresponding sides of a similar triangle.
So, the only option that works is option 2. The measures of the corresponding sides of the similar triangle could be 9 inches and 12 inches.