Final answer;
The length of
is
(Option B). This is determined using the Triangle Proportionality Theorem, with the given information about the length of
and the parallel lines formed within the triangle.Thus the correct option is:b. 32 units
Step-by-step explanation:
In the given triangle
let \( G \) be the point on
and
be the point on
such that
is marked with a length of 16 units.
To find the length of
we can use the Triangle Proportionality Theorem, which states that if a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides proportionally.
Let
be the point on
According to the Triangle Proportionality Theorem, we have:
![\[ \frac{\overline{AG}}{\overline{GB}} = \frac{\overline{AI}}{\overline{IC}} \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/exu0163d6i6axg37brhdszvv3lf9m1ygej.png)
Since
and
we can set up the proportion:
![\[ \frac{\overline{AG}}{\overline{AB}} = \frac{\overline{AI}}{\overline{AC}} \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/v2qlxxxygqkr486u9ua4vthe88chpl3a4n.png)
Given that
units, we can express
as
Substituting this into the proportion, we get:
![\[ \frac{\overline{AB} - \overline{GB}}{\overline{AB}} = \frac{\overline{AI}}{\overline{AC}} \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/xk16o0jbamtpxnbvv944vr8g8ik80og0tq.png)
Now, solving for

![\[ \overline{AC} = \frac{\overline{AI} \cdot \overline{AB}}{\overline{AB} - \overline{GB}} \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/98pzg575amh3matfirfux65jxh7syjf0uv.png)
Finally, substituting the given values, we find:
![\[ \overline{AC} = \frac{16 \cdot \overline{AB}}{\overline{AB} - 16} \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/5jd17b3p0a1ujhevqklapf5js02zoki3l1.png)
![\[ \overline{AC} = (16 \cdot 48)/(48 - 16) \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/86taft7vgdnp5siabikcozde09mj8npomb.png)
![\[ \overline{AC} = (768)/(32) \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/msbdnvewxrbsr79jiazeiucx3wuyh3sk8l.png)
![\[ \overline{AC} = 24 \text{ units} \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/zqt4e0nwmpup50ebsfi68g0x8p35l76j88.png)
Therefore, the length of
is 24 units, and the correct answer is Option B.