Question
Regina draws a triangle inside a circle; she labels the vertices ABC , with the diameter as AB and the inscribed angle as C . She draws the length of AB 14 cm long, and AC 8 cm long. Using what you know about inscribed angles, what will be the approximate angle measure of B ?(1 point) Responses 90° 90° 55° 55° 70° 70° 35°
Answers
GPT-4o mini
In this scenario, since \(AB\) is the diameter of the circle and point \(C\) is on the circle, angle \(C\) is an inscribed angle that intercepts the arc subtended by diameter \(AB\). According to the inscribed angle theorem, we know that the inscribed angle that subtends a diameter is a right angle. Therefore, angle \(C\) is exactly \(90^\circ\).
Now, using the triangle angle sum property, we know that the angles in triangle \(ABC\) must sum to \(180^\circ\):
\[
A + B + C = 180^\circ
\]
Substituting \(C = 90^\circ\) into the equation:
\[
A + B + 90^\circ = 180^\circ
\]
This reduces to:
\[
A + B = 90^\circ
\]
Now, we need to understand the lengths \(AC\) and \(AB\) to find angle \(B\). Since \(AB\) is the diameter and equal to \(14 \, cm\), we can utilize the triangle properties and the fact that we also have side \(AC = 8 \, cm\).
We can apply the Law of Sines in triangle \(ABC\):
\[
\frac{AC}{\sin(B)} = \frac{AB}{\sin(C)}
\]
Substituting known values, we have:
\[
\frac{8}{\sin(B)} = \frac{14}{\sin(90^\circ)}
\]
Since \(\sin(90^\circ) = 1\), the equation simplifies to:
\[
\frac{8}{\sin(B)} = 14
\]
Cross-multiplying gives:
\[
8 = 14 \sin(B)
\]
Solving for \(\sin(B)\):
\[
\sin(B) = \frac{8}{14} = \frac{4}{7}
\]
Next, we find angle \(B\) using the inverse sine function:
\[
B \approx \sin^{-1}\left(\frac{4}{7}\right)
\]
Calculating that gives approximately:
\[
B \approx 45.6^\circ
\]
Since that value doesn't appear in the options provided, we re-evaluate the triangle's dimensions in terms of possible angles based on the principle of triangle properties.
Since we derived that \(C = 90^\circ \), and \(A + B = 90^\circ\), \(B\) must be a complementary angle to \(A\). Using estimation, the closeness to intersecting angle possibilities of triangle configurations suggests \(B\) being less than or equal to \(55^\circ\).
Therefore, the most appropriate and closest choice from your options would actually lead us to respond with:
**55°**
This is an estimation derived from potential angles naturally forming in a right-triangle configuration provided the given segments.
Now, using the triangle angle sum property, we know that the angles in triangle \(ABC\) must sum to \(180^\circ\):
\[
A + B + C = 180^\circ
\]
Substituting \(C = 90^\circ\) into the equation:
\[
A + B + 90^\circ = 180^\circ
\]
This reduces to:
\[
A + B = 90^\circ
\]
Now, we need to understand the lengths \(AC\) and \(AB\) to find angle \(B\). Since \(AB\) is the diameter and equal to \(14 \, cm\), we can utilize the triangle properties and the fact that we also have side \(AC = 8 \, cm\).
We can apply the Law of Sines in triangle \(ABC\):
\[
\frac{AC}{\sin(B)} = \frac{AB}{\sin(C)}
\]
Substituting known values, we have:
\[
\frac{8}{\sin(B)} = \frac{14}{\sin(90^\circ)}
\]
Since \(\sin(90^\circ) = 1\), the equation simplifies to:
\[
\frac{8}{\sin(B)} = 14
\]
Cross-multiplying gives:
\[
8 = 14 \sin(B)
\]
Solving for \(\sin(B)\):
\[
\sin(B) = \frac{8}{14} = \frac{4}{7}
\]
Next, we find angle \(B\) using the inverse sine function:
\[
B \approx \sin^{-1}\left(\frac{4}{7}\right)
\]
Calculating that gives approximately:
\[
B \approx 45.6^\circ
\]
Since that value doesn't appear in the options provided, we re-evaluate the triangle's dimensions in terms of possible angles based on the principle of triangle properties.
Since we derived that \(C = 90^\circ \), and \(A + B = 90^\circ\), \(B\) must be a complementary angle to \(A\). Using estimation, the closeness to intersecting angle possibilities of triangle configurations suggests \(B\) being less than or equal to \(55^\circ\).
Therefore, the most appropriate and closest choice from your options would actually lead us to respond with:
**55°**
This is an estimation derived from potential angles naturally forming in a right-triangle configuration provided the given segments.