Question

A circle is shown. Secant A D and tangent E D intersect at point D outside of the circle. Secant A D intersects the circle at point B. The length of A B is a, the length of B D is 10, and the length of D E is 12.
Which equation results from applying the secant and tangent segment theorem to the figure?

12(a + 12) = 102
10 + 12 = a2
10(a + 10) = 122
10(12) = a2

Answers

Answered by GPT-4o mini
The secant and tangent segment theorem states that if a tangent segment and a secant segment intersect at a point outside of the circle, then the square of the length of the tangent segment is equal to the product of the lengths of the entire secant segment and its external segment.

In this case, the tangent segment is \( D E \) and has a length of \( 12 \). The secant segment is \( A D \), which extends from point \( A \) to point \( D \) and intersects the circle at point \( B \). According to the information given:

- Length of \( A B = a \)
- Length of \( B D = 10 \)

Thus, the entire length of the secant segment \( A D \) would be:

\[
A D = A B + B D = a + 10
\]

According to the theorem:

\[
\text{(Tangent Segment)}^2 = \text{(Secant Segment)} \times \text{(External Segment)}
\]

So we have:

\[
D E^2 = A D \cdot B D
\]

Substituting the lengths we have:

\[
12^2 = (a + 10) \cdot 10
\]

Calculating \( 12^2 \):

\[
144 = 10(a + 10)
\]

We can simplify this to:

\[
144 = 10a + 100
\]

Now, this can be rewritten as:

\[
10a = 144 - 100
\]
\[
10a = 44
\]
\[
a = 4.4
\]

This means the result from applying the secant and tangent segment theorem is:

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

However, out of the options given, the best match for our derived equation is \( 10(a + 10) = 122 \) (if 122 implies computed value of \( 12^2 \) as \( 144 \), which must refer to the total equation form as stated).

From evaluating the context, thus:

\[
\text{The equation that results from applying the secant and tangent segment theorem is: } 10(a + 10) = 122.
\]

So the correct equation to choose is:
**10(a + 10) = 122**

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