Question

Triangle ABC has side lengths of 10, 12, and 8 inches. Suppose you want to draw a congruent and smaller triangle within triangle ABC. What is a possible midsegment length of the smaller triangle?

Answer 1

To find a possible midsegment length of a smaller triangle drawn within triangle ABC, we first need to understand what a midsegment is. A midsegment of a triangle is a 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.

Given the side lengths of triangle ABC as 10 inches, 12 inches, and 8 inches, we can identify the sides as follows:

- Let \( AB = 10 \) inches
- Let \( BC = 12 \) inches
- Let \( AC = 8 \) inches

If we choose to draw a midsegment that connects the midpoints of two sides, the midsegment will be half the length of the side that it is parallel to.

### Calculating the Midsegments

1. **If the midsegment is parallel to side BC (12 inches):**
\[
\text{Midsegment length} = \frac{1}{2} \times 12 = 6 \text{ inches}
\]

2. **If the midsegment is parallel to side AC (8 inches):**
\[
\text{Midsegment length} = \frac{1}{2} \times 8 = 4 \text{ inches}
\]

3. **If the midsegment is parallel to side AB (10 inches):**
\[
\text{Midsegment length} = \frac{1}{2} \times 10 = 5 \text{ inches}
\]

### Conclusion
Therefore, the possible midsegment lengths of a smaller triangle drawn within triangle ABC could be \( 6 \) inches, \( 4 \) inches, or \( 5 \) inches. These lengths correspond to the midsegments that are parallel to the sides of triangle ABC.