Question

what is the length of \\( \overline{sa} \\)?
\\( \bigcirc 1.89 \mathrm{ft} \\)
\\( \bigcirc 2.43 \mathrm{ft} \\)
\\( \bigcirc 3 \mathrm{ft} \\)
\\( \bigcirc 7 \mathrm{ft} \\)

Explanation:

Step1: Use the property of similar triangles

Since \(ST\parallel AC\), \(\triangle BST\sim\triangle BAC\).

Step2: Set up the proportion

By the property of similar triangles, \(\frac{BS}{BA}=\frac{BT}{BC}\). Let \(SA = x\), then \(BA=x + 10\), \(BT = 9\), \(BC=9 + 2.7=11.7\). But another way: using the mid - segment theorem (if \(ST\) is parallel to \(AC\) and \(T\) divides \(BC\) proportionally). Also, we can use the formula \(\frac{SA}{BS}=\frac{CT}{BT}\).

Step3: Substitute the values

We know \(BS = 10\), \(BT=9\), \(CT = 2.7\). Let \(SA=x\). From \(\frac{x}{10}=\frac{2.7}{6.3}\) (using the basic proportionality theorem: if a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides proportionally). Cross - multiply: \(6.3x=10\times2.7\).

Step4: Solve for \(x\)

\(x=\frac{10\times2.7}{6.3}=\frac{27}{6.3}=\frac{270}{63}=\frac{30}{7}\approx7\) (after simplifying \(\frac{27}{6.3}=\frac{270}{63}=\frac{30}{7}\approx7\)).

Answer:

\(7\mathrm{ft}\)