Question

what is the length of \\(\overline{sa}\\)?

1.89 ft
2.43 ft
3 ft
7 ft

Explanation:

Identify parallel lines and similar triangles

Using the Triangle Geometry knowledge point, we observe that segment \(ST\) is parallel to segment \(AC\) because of the matching parallel arrows on the diagram:

$$ST \parallel AC$$

Since \(ST \parallel AC\), the corresponding angles are equal:

$$\angle BST = \angle BAC$$

$$\angle BTS = \angle BCA$$

Therefore, by Angle-Angle (AA) similarity, triangle \(BST\) is similar to triangle \(BAC\):

$$\triangle BST \sim \triangle BAC$$

Set up the ratio of corresponding sides

Since \(\triangle BST \sim \triangle BAC\), the ratios of their corresponding side lengths are equal:

$$\frac{BS}{BA} = \frac{BT}{BC}$$

From the diagram, we are given the following lengths:

$$BS = 10\text{ ft}$$

$$BT = 9\text{ ft}$$

$$TC = 2.7\text{ ft}$$

The total length of segment \(BC\) is:

$$BC = BT + TC = 9 + 2.7 = 11.7\text{ ft}$$

Let the length of segment \(SA\) be \(x\). The total length of segment \(BA\) is:

$$BA = BS + SA = 10 + x$$

Solve for the unknown length

Substitute the known values into the similarity ratio:

$$\frac{10}{10 + x} = \frac{9}{11.7}$$

Cross-multiply to solve for \(x\):

$$10 \cdot 11.7 = 9(10 + x)$$

$$117 = 90 + 9x$$

$$9x = 27$$

$$x = 3\text{ ft}$$

Thus, the length of \(\overline{SA}\) is \(3\text{ ft}\).

Answer:

• 1.89 ft
• 2.43 ft
• 3 ft (Correct answer)
• 7 ft