To solve for \( CB \) in the described figure with provided information, we will utilize geometric properties involving tangents and circles.
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Understanding the Given Information:
- We have a circle with center \( O \).
- A tangent line at point \( A \) creates angle \( \angle ABD = 38^\circ \) where \( B \) is a point outside the circle and \( D \) is another point on the tangent line.
- The radius \( OA \) is given as \( 12.6 \) meters.
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Geometric Relationships:
- By the properties of tangents, the radius drawn to the point of tangency \( A \) is perpendicular to the tangent line \( AB \).
- Thus, \( \angle OAD = 90^\circ \).
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Finding Angles:
- By the exterior angle theorem in triangle \( OAB \): \[ \angle OAB + \angle ABD = \angle OAD \] \[ \angle OAB + 38^\circ = 90^\circ \] \[ \angle OAB = 90^\circ - 38^\circ = 52^\circ \]
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Using Trigonometry:
- In triangle \( OAB \), we can use trigonometric functions to find \( AB \):
\[
\tan(\angle OAB) = \frac{OA}{AB}
\]
- Here, \( OA = 12.6 \) meters. \[ \tan(52^\circ) = \frac{12.6}{AB} \] \[ AB = \frac{12.6}{\tan(52^\circ)} \]
- Calculate \( \tan(52^\circ) \approx 1.2799 \): \[ AB \approx \frac{12.6}{1.2799} \approx 9.85 , \text{meters} \]
- In triangle \( OAB \), we can use trigonometric functions to find \( AB \):
\[
\tan(\angle OAB) = \frac{OA}{AB}
\]
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Finding \( CB \):
- Since \( CB \) can be calculated by knowing \( AB \) and any other relationship in the triangle or line we might have (assuming, for example, there was another triangle present or a specific connector between \( C \) and \( B \); here not enough information was specified for \( C \)).
If we lack sufficient details for \( C \) or the length \( AC \) or any reference to \( C \), we cannot directly calculate \( CB \) from the given information with certainty.
Conclusion
Assuming that you wanted just \( AB \):
- The length \( AB \approx 9.85 , \text{meters} \).
For \( CB \) specifically, please provide additional context or constraints so we can derive that length as well.
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