Question
In circle NN, m, angle, O, N, P, equals, 48, degreesm∠ONP=48
∘
. Solve for xx if m, arc, O, P, equals, left parenthesis, 4, x, minus, 48, right parenthesis, degreesm
OP
⌢
=(4x−48)
∘
. If necessary, round your answer to the nearest tenth.
∘
. Solve for xx if m, arc, O, P, equals, left parenthesis, 4, x, minus, 48, right parenthesis, degreesm
OP
⌢
=(4x−48)
∘
. If necessary, round your answer to the nearest tenth.
Answers
GPT-4o mini
In a circle, the measure of the angle formed by two lines radiating from a common point (the vertex of the angle) is half the measure of the arc that the angle subtends.
In this case, we have the following:
- \( m\angle ONP = 48^\circ \)
- \( m\arc{OP} = (4x - 48)^\circ \)
According to the relationship between the angle and the arc, we have:
\[
m\angle ONP = \frac{1}{2} m\arc{OP}
\]
Substituting the values we have:
\[
48 = \frac{1}{2} (4x - 48)
\]
Now, we'll solve for \( x \):
1. Multiply both sides by 2 to eliminate the fraction:
\[
96 = 4x - 48
\]
2. Add 48 to both sides:
\[
96 + 48 = 4x
\]
\[
144 = 4x
\]
3. Divide both sides by 4:
\[
x = \frac{144}{4}
\]
\[
x = 36
\]
Thus, the value of \( x \) is \( \boxed{36} \).
In this case, we have the following:
- \( m\angle ONP = 48^\circ \)
- \( m\arc{OP} = (4x - 48)^\circ \)
According to the relationship between the angle and the arc, we have:
\[
m\angle ONP = \frac{1}{2} m\arc{OP}
\]
Substituting the values we have:
\[
48 = \frac{1}{2} (4x - 48)
\]
Now, we'll solve for \( x \):
1. Multiply both sides by 2 to eliminate the fraction:
\[
96 = 4x - 48
\]
2. Add 48 to both sides:
\[
96 + 48 = 4x
\]
\[
144 = 4x
\]
3. Divide both sides by 4:
\[
x = \frac{144}{4}
\]
\[
x = 36
\]
Thus, the value of \( x \) is \( \boxed{36} \).