Question
In the figure below, line LN is tangent to circle O.
Point M is at the center of the tangent L-N, the points O-M are joined, and points O-N are joined.
Which of the following can be concluded from the information?
A.
¯¯¯¯¯¯
O
N
is a radius of the circle.
B.
△
M
N
O
is a right triangle.
C.
∠
O
N
M
is an obtuse angle.
D.
∠
O
M
N
is a central angle of circle O.
Point M is at the center of the tangent L-N, the points O-M are joined, and points O-N are joined.
Which of the following can be concluded from the information?
A.
¯¯¯¯¯¯
O
N
is a radius of the circle.
B.
△
M
N
O
is a right triangle.
C.
∠
O
N
M
is an obtuse angle.
D.
∠
O
M
N
is a central angle of circle O.
Answers
Answered by
GPT-4o mini
To analyze the given information, let's break down each statement in relation to the setup involving a circle with center O and a tangent line LN touching the circle at point N.
1. **Statement A: \( \overline{ON} \) is a radius of the circle.**
- This statement is true because N is the point where the tangent line LN touches the circle, and therefore the line segment from O (the center) to N (the point of tangency) represents a radius.
2. **Statement B: \( \triangle MNO \) is a right triangle.**
- This statement is also true. Since line LN is tangent to the circle at point N, the radius \( \overline{ON} \) is perpendicular to the tangent line LN at point N. Thus, triangle MNO, with MN being a tangent from M to N, will have right angle ∠MNO.
3. **Statement C: \( \angle ONM \) is an obtuse angle.**
- This statement is false. Since \( \angle ONM \) is part of triangle MNO, and we know one angle (∠MNO) is a right angle, the remaining angles must sum to 90 degrees. Therefore, \( \angle ONM \) cannot be obtuse.
4. **Statement D: \( \angle OMN \) is a central angle of circle O.**
- This statement is false. A central angle is one where the vertex is the center of the circle and the sides of the angle go to points on the circle. Here, the angle \( \angle OMN \) has point M (not O) as its vertex, so it cannot be a central angle.
Based on this analysis, the conclusions are:
- Statement A is true.
- Statement B is true.
- Statement C is false.
- Statement D is false.
From the options, the ones that can be concluded from the information given are A and B. If you have to choose just one, A would be a fundamental property of circles and the concept of tangents, making it a stronger candidate for a conclusion.
1. **Statement A: \( \overline{ON} \) is a radius of the circle.**
- This statement is true because N is the point where the tangent line LN touches the circle, and therefore the line segment from O (the center) to N (the point of tangency) represents a radius.
2. **Statement B: \( \triangle MNO \) is a right triangle.**
- This statement is also true. Since line LN is tangent to the circle at point N, the radius \( \overline{ON} \) is perpendicular to the tangent line LN at point N. Thus, triangle MNO, with MN being a tangent from M to N, will have right angle ∠MNO.
3. **Statement C: \( \angle ONM \) is an obtuse angle.**
- This statement is false. Since \( \angle ONM \) is part of triangle MNO, and we know one angle (∠MNO) is a right angle, the remaining angles must sum to 90 degrees. Therefore, \( \angle ONM \) cannot be obtuse.
4. **Statement D: \( \angle OMN \) is a central angle of circle O.**
- This statement is false. A central angle is one where the vertex is the center of the circle and the sides of the angle go to points on the circle. Here, the angle \( \angle OMN \) has point M (not O) as its vertex, so it cannot be a central angle.
Based on this analysis, the conclusions are:
- Statement A is true.
- Statement B is true.
- Statement C is false.
- Statement D is false.
From the options, the ones that can be concluded from the information given are A and B. If you have to choose just one, A would be a fundamental property of circles and the concept of tangents, making it a stronger candidate for a conclusion.