Question
In circle PP, m, angle, N, O, Q, equals, 62, degreesm∠NOQ=62
∘
. Solve for xx if m, arc, N, Q, equals, left parenthesis, 3, x, minus, 31, right parenthesis, degreesm
NQ
⌢
=(3x−31)
∘
. If necessary, round your answer to the nearest tenth.
∘
. Solve for xx if m, arc, N, Q, equals, left parenthesis, 3, x, minus, 31, right parenthesis, degreesm
NQ
⌢
=(3x−31)
∘
. If necessary, round your answer to the nearest tenth.
Answers
GPT-4o mini
To solve for \(x\) given the angle \( m \angle NOQ = 62^\circ \) and the arc \( m \arc{NQ} = (3x - 31)^\circ \), we can use the relationship between the angle and the arc in a circle.
In a circle, the measure of an inscribed angle is half the measure of the arc it intercepts. Therefore, we have:
\[
m \angle NOQ = \frac{1}{2} m \arc{NQ}
\]
Substituting the given values:
\[
62^\circ = \frac{1}{2} (3x - 31)
\]
Now, we will solve for \(x\). First, multiply both sides of the equation by 2 to eliminate the fraction:
\[
2 \cdot 62^\circ = 3x - 31
\]
This simplifies to:
\[
124 = 3x - 31
\]
Next, add 31 to both sides to isolate the term with \(x\):
\[
124 + 31 = 3x
\]
\[
155 = 3x
\]
Now, divide both sides by 3 to solve for \(x\):
\[
x = \frac{155}{3} \approx 51.67
\]
Rounding to the nearest tenth, we find:
\[
x \approx 51.7
\]
Therefore, the solution is:
\[
\boxed{51.7}
\]
In a circle, the measure of an inscribed angle is half the measure of the arc it intercepts. Therefore, we have:
\[
m \angle NOQ = \frac{1}{2} m \arc{NQ}
\]
Substituting the given values:
\[
62^\circ = \frac{1}{2} (3x - 31)
\]
Now, we will solve for \(x\). First, multiply both sides of the equation by 2 to eliminate the fraction:
\[
2 \cdot 62^\circ = 3x - 31
\]
This simplifies to:
\[
124 = 3x - 31
\]
Next, add 31 to both sides to isolate the term with \(x\):
\[
124 + 31 = 3x
\]
\[
155 = 3x
\]
Now, divide both sides by 3 to solve for \(x\):
\[
x = \frac{155}{3} \approx 51.67
\]
Rounding to the nearest tenth, we find:
\[
x \approx 51.7
\]
Therefore, the solution is:
\[
\boxed{51.7}
\]