Question

which equation shows the correct application of the secant-tangent segment theorem to the figure?

\\(12(a + 12) = 10^2\\)
\\(10 + 12 = a^2\\)
\\(10(a + 10) = 12^2\\)
\\(10(12) = a^2\\)

Explanation:

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<reasoning>

Identify the given segments

\[

$$\begin{aligned} &\text{External secant segment} = 10\\ &\text{Internal secant segment (chord)} = a\\ &\text{Entire secant segment} = 10 + a\\ &\text{Tangent segment} = 12 \end{aligned}$$

\]

Apply the tangent-secant theorem

\[

$$\begin{aligned} &\text{External Secant} \times \text{Entire Secant} = (\text{Tangent})^2\\ &10(a + 10) = 12^2 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>\(12(a + 12) = 10^2\)</mcq-option>
<mcq-option>\(10 + 12 = a^2\)</mcq-option>
<mcq-correct>\(10(a + 10) = 12^2\)</mcq-correct>
<mcq-option>\(10(12) = a^2\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the given segments

\[

$$\begin{aligned} &\text{External secant segment} = 10\\ &\text{Internal secant segment (chord)} = a\\ &\text{Entire secant segment} = 10 + a\\ &\text{Tangent segment} = 12 \end{aligned}$$

\]

Apply the tangent-secant theorem

\[

$$\begin{aligned} &\text{External Secant} \times \text{Entire Secant} = (\text{Tangent})^2\\ &10(a + 10) = 12^2 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>\(12(a + 12) = 10^2\)</mcq-option>
<mcq-option>\(10 + 12 = a^2\)</mcq-option>
<mcq-correct>\(10(a + 10) = 12^2\)</mcq-correct>
<mcq-option>\(10(12) = a^2\)</mcq-option>
</answer>

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