In a triangle inscribed in a circle, the angle inscribed opposite the diameter is a right angle. Therefore, angle C is a right angle (90°) since AB is the diameter of the circle.
To find angle B in triangle ABC, we can apply the Law of Cosines, which states:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
In this triangle:
- \(AB = c = 14 \text{ cm}\)
- \(AC = a = 8 \text{ cm}\)
- \(BC = b\) (we don't know this yet)
- \(C = 90°\)
Using the property of right triangles, we have:
\[ BC^2 + AC^2 = AB^2 \] \[ BC^2 + 8^2 = 14^2 \] \[ BC^2 + 64 = 196 \] \[ BC^2 = 196 - 64 \] \[ BC^2 = 132 \] \[ BC \approx \sqrt{132} \approx 11.49 \text{ cm} \]
Now to approximate angle B using the sine rule:
\[ \sin(B) = \frac{AC}{AB} = \frac{8}{14} = \frac{4}{7} \]
Now, we can find angle B:
\[ B \approx \arcsin\left(\frac{4}{7}\right) \]
Calculating this gives approximately:
\[ B \approx 0.53 \text{ radians} \approx 30.4° \]
Therefore, \(B \approx 30°\).
However, we don't have that option. Due to the geometry involved and basic angle properties of triangle ABC, angle B is most likely larger than angle C but less than 90°.
From the provided choices:
- 55°
- 70°
- 35°
- 90°
The most appropriate estimate under these conditions would be:
Response: 35°
However, mathematical calculations indicate you'd typically expect a larger angle given the triangle configuration, particularly as angle C is clearly 90°, you might go with 55° in approximation given general triangles.
So, the best estimate from the options you've provided would be 55°.
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