Question

Consider ∆abc. a triangle a b c with points g, and h marked between a b, and b c. the length of g h is 16. what is the length of ac? a. 48 units b. 32 units c. 24 units d. 16 units

Answer 1

Final answer;

The length of
`\( \overline{AC} \)` is
`\( \boxed{\text{32 units}} \)` (Option B). This is determined using the Triangle Proportionality Theorem, with the given information about the length of
`\( \overline{GH} \)`and the parallel lines formed within the triangle.Thus the correct option is:b. 32 units

Step-by-step explanation:

In the given triangle
`\( \triangle ABC \),` let \( G \) be the point on
`\( \overline{AB} \)`and
`\( H \)`be the point on
`\( \overline{BC} \),` such that
`\( \overline{GH} \)` is marked with a length of 16 units.

To find the length of
`\( \overline{AC} \),` 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
`\( I \)` be the point on
`\( \overline{AC} \) such that \( \overline{GH} \parallel \overline{IB} \).` According to the Triangle Proportionality Theorem, we have:

`\[ \frac{\overline{AG}}{\overline{GB}} = \frac{\overline{AI}}{\overline{IC}} \]`

Since
`\( \overline{AG} + \overline{GB} = \overline{AB} \),` and
`\( \overline{AI} + \overline{IC} = \overline{AC} \),` we can set up the proportion:

`\[ \frac{\overline{AG}}{\overline{AB}} = \frac{\overline{AI}}{\overline{AC}} \]`

Given that
`\( \overline{GH} = 16 \)` units, we can express
`\( \overline{AG} \)`as
`\( \overline{AB} - \overline{GB} \).` Substituting this into the proportion, we get:

`\[ \frac{\overline{AB} - \overline{GB}}{\overline{AB}} = \frac{\overline{AI}}{\overline{AC}} \]`

Now, solving for
`\( \overline{AC} \):`

`\[ \overline{AC} = \frac{\overline{AI} \cdot \overline{AB}}{\overline{AB} - \overline{GB}} \]`

Finally, substituting the given values, we find:

`\[ \overline{AC} = \frac{16 \cdot \overline{AB}}{\overline{AB} - 16} \]`

`\[ \overline{AC} = (16 \cdot 48)/(48 - 16) \]`

`\[ \overline{AC} = (768)/(32) \]`

`\[ \overline{AC} = 24 \text{ units} \]`

Therefore, the length of
`\( \overline{AC} \)`is 24 units, and the correct answer is Option B.

Answer 2

Without sufficient information, it is impossible to determine the length of AC based on the length of GH alone.

Step-by-step explanation:

To determine the length of side AC in △ABC, it is necessary to understand the relationships within the triangle, and it seems that the question implies that there could be a proportional relationship or a specific geometric principle at play. Unfortunately, the information provided in the question is not sufficient to determine the length of AC, as there is no clear relationship given between GH and AC, nor is there any additional information about angles or other sides that could be used with trigonometric or Pythagorean principles. More information is needed to provide an accurate answer to the student's question.