Question
Line segment SU is a diameter of circle V.
Circle V is inscribed with triangle S T U. Line S U is a diameter. Lines S T and T U are secants. Angle S T U intercepts arc T U and has a measure of 34 degrees.
What is the measure of arc S T?
56°
68°
112°
163°
Circle V is inscribed with triangle S T U. Line S U is a diameter. Lines S T and T U are secants. Angle S T U intercepts arc T U and has a measure of 34 degrees.
What is the measure of arc S T?
56°
68°
112°
163°
Answers
Answered by
GPT-4o mini
To find the measure of arc ST, we'll use the relationship between an angle inscribed in a circle and the arcs it intercepts.
Given that angle \( STU \) intercepts arc \( TU \), the measure of the angle \( \angle STU \) is half the measure of the arc \( TU \):
\[
\angle STU = \frac{1}{2} \cdot m(arc \, TU)
\]
We know from the problem that \( \angle STU = 34^\circ \).
To find the measure of arc \( TU \):
\[
34^\circ = \frac{1}{2} \cdot m(arc \, TU)
\]
To solve for \( m(arc \, TU) \):
\[
m(arc \, TU) = 2 \cdot 34^\circ = 68^\circ
\]
Next, we also know that in a circle, the sum of the measures of the arcs \( ST \), \( TU \), and \( US \) equals 360 degrees. Since \( SU \) is a diameter of circle \( V \), arc \( SU \) (which is the same as arc \( UT \) going the other way) measures 180 degrees.
Thus, we can express the total arcs in the circle:
\[
m(arc \, ST) + m(arc \, TU) + m(arc \, SU) = 360^\circ
\]
Now we can substitute \( m(arc \, TU) = 68^\circ \) and \( m(arc \, SU) = 180^\circ \):
\[
m(arc \, ST) + 68^\circ + 180^\circ = 360^\circ
\]
Combining the constants gives us:
\[
m(arc \, ST) + 248^\circ = 360^\circ
\]
Subtracting 248 from both sides gives us:
\[
m(arc \, ST) = 360^\circ - 248^\circ = 112^\circ
\]
Therefore, the measure of arc \( ST \) is \( \boxed{112^\circ} \).
Given that angle \( STU \) intercepts arc \( TU \), the measure of the angle \( \angle STU \) is half the measure of the arc \( TU \):
\[
\angle STU = \frac{1}{2} \cdot m(arc \, TU)
\]
We know from the problem that \( \angle STU = 34^\circ \).
To find the measure of arc \( TU \):
\[
34^\circ = \frac{1}{2} \cdot m(arc \, TU)
\]
To solve for \( m(arc \, TU) \):
\[
m(arc \, TU) = 2 \cdot 34^\circ = 68^\circ
\]
Next, we also know that in a circle, the sum of the measures of the arcs \( ST \), \( TU \), and \( US \) equals 360 degrees. Since \( SU \) is a diameter of circle \( V \), arc \( SU \) (which is the same as arc \( UT \) going the other way) measures 180 degrees.
Thus, we can express the total arcs in the circle:
\[
m(arc \, ST) + m(arc \, TU) + m(arc \, SU) = 360^\circ
\]
Now we can substitute \( m(arc \, TU) = 68^\circ \) and \( m(arc \, SU) = 180^\circ \):
\[
m(arc \, ST) + 68^\circ + 180^\circ = 360^\circ
\]
Combining the constants gives us:
\[
m(arc \, ST) + 248^\circ = 360^\circ
\]
Subtracting 248 from both sides gives us:
\[
m(arc \, ST) = 360^\circ - 248^\circ = 112^\circ
\]
Therefore, the measure of arc \( ST \) is \( \boxed{112^\circ} \).