△ABC has side lengths of 10, 16, and 23 units. Suppose you want to draw a congruent and smaller triangle within ΔABC. Investigate a pattern between the original triangle and smaller triangle and make a conjecture. What is a possible midsegment length of the smaller triangle? (1 point)

Question

Responses

4 units
4 units

10 units
10 units

8 units
8 units

13 units

Answer 1

To determine a possible midsegment length of a smaller triangle that is congruent to triangle \( \triangle ABC \) with side lengths of 10, 16, and 23 units, we can use the concept of the midsegment of a triangle.

The midsegment of a triangle connects the midpoints of two sides. The length of the midsegment is half the length of the side it is parallel to. Therefore, if we find the midsegments of triangle \( \triangle ABC \), we can draw a smaller triangle inside it.

To find the lengths of the midsegments of \( \triangle ABC \):

1. Midsegment parallel to side of length 10:

   • Length = \( \frac{10}{2} = 5 \) units

2. Midsegment parallel to side of length 16:

   • Length = \( \frac{16}{2} = 8 \) units

3. Midsegment parallel to side of length 23:

   • Length = \( \frac{23}{2} = 11.5 \) units

Given the possible responses (4, 10, 8, and 13 units), the only midsegment length that appears as a possibility from our calculations is 8 units.

Thus, the conjecture is that if you create a smaller triangle by connecting midpoints of \( \triangle ABC \), the length of the midsegment will be proportional, and one possible midsegment length of the smaller triangle can indeed be 8 units.