To find the midsegment lengths of the smaller triangle that is congruent to triangle ABC, we need to remember that a midsegment of a triangle is a line segment that connects the midpoints of two sides of the triangle. The midsegment is parallel to the third side and its length is half the length of that side.
In triangle ABC with sides of lengths 10, 12, and 8 inches, we can calculate the lengths of the midsegments:
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The midsegment corresponding to side 10 inches will have a length of: \[\text{Midsegment} = \frac{10}{2} = 5 \text{ inches}\]
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The midsegment corresponding to side 12 inches will have a length of: \[\text{Midsegment} = \frac{12}{2} = 6 \text{ inches}\]
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The midsegment corresponding to side 8 inches will have a length of: \[\text{Midsegment} = \frac{8}{2} = 4 \text{ inches}\]
Since we are trying to find a length that is possible for a midsegment of the smaller triangle, we can state the midsegment lengths as follows:
- Midsegment related to side 10: 5 inches
- Midsegment related to side 12: 6 inches
- Midsegment related to side 8: 4 inches
From the options provided (8 inches, 6 inches, 10 inches, 16 inches), the only possible midsegment length for a smaller triangle, which is parallel to one of the sides of triangle ABC and fulfills the condition of being half the length of one of the triangle's sides, is:
6 inches
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