Answer :
To solve this problem, we need to understand the relationship between a secant and a tangent meeting at a 90° angle outside the circle and the measures of the intercepted arcs.
### Step-by-Step Solution:
1. Conceptual Understanding:
- When a secant and a tangent meet at a point outside the circle, they form an angle.
- Specifically, in this case, the angle formed is given as 90°.
2. Circle Theorems:
- According to circle theorems, when a secant (line that intersects the circle at two points) and a tangent (line that touches the circle at exactly one point) meet outside the circle and form a 90° angle, there is a specific relationship between the intercepted arcs.
- The particular theorem we use here states that the difference between the measures of the two intercepted arcs (the major arc and the minor arc on either side of the secant) is always equal to 180°.
3. Application:
- Given that the tangent and secant meet at a 90° angle, we apply this theorem directly.
- Therefore, the difference between the measures of the intercepted arcs when they intersect at this angle is found to be 180°.
4. Conclusion:
- Based on the above understanding, the difference between the measures of the intercepted arcs is determined to be 180°.
Thus, the correct answer is 180°.
### Step-by-Step Solution:
1. Conceptual Understanding:
- When a secant and a tangent meet at a point outside the circle, they form an angle.
- Specifically, in this case, the angle formed is given as 90°.
2. Circle Theorems:
- According to circle theorems, when a secant (line that intersects the circle at two points) and a tangent (line that touches the circle at exactly one point) meet outside the circle and form a 90° angle, there is a specific relationship between the intercepted arcs.
- The particular theorem we use here states that the difference between the measures of the two intercepted arcs (the major arc and the minor arc on either side of the secant) is always equal to 180°.
3. Application:
- Given that the tangent and secant meet at a 90° angle, we apply this theorem directly.
- Therefore, the difference between the measures of the intercepted arcs when they intersect at this angle is found to be 180°.
4. Conclusion:
- Based on the above understanding, the difference between the measures of the intercepted arcs is determined to be 180°.
Thus, the correct answer is 180°.
