A secant and a tangent meet at a 90° angle outside the circle. What must be the difference between the measures of the intercepted arcs?
A. 45°
B. 90°
C. 180°
D. 270°
Answer (1)
Define the larger and smaller intercepted arcs as x and y , respectively.
Express the relationship between the angle formed by the secant and tangent and the intercepted arcs: 90 = 2 1 ( x − y ) .
Solve the equation for the difference between the arcs: x − y = 180 .
State the final answer: 18 0 ∘ .
Explanation
Problem Analysis Let's analyze the problem. We have a secant and a tangent intersecting outside a circle at a 90° angle. We need to find the difference between the measures of the intercepted arcs.
Set up the equation Let the measure of the larger intercepted arc be x and the measure of the smaller intercepted arc be y . The angle formed by the secant and tangent is half the difference of the intercepted arcs. Therefore, we can write the equation: 90 = 2 1 ( x − y )
Solve for the difference Now, let's solve the equation for x − y . Multiply both sides of the equation by 2: 2 × 90 = 2 × 2 1 ( x − y ) 180 = x − y So, the difference between the measures of the intercepted arcs is 180 degrees.
Final Answer Therefore, the difference between the measures of the intercepted arcs is 18 0 ∘ .
Examples
Understanding angles formed by secants and tangents is crucial in fields like astronomy, where determining the positions of celestial bodies involves analyzing angles formed by lines of sight (tangents) and paths of objects (secants) relative to a circular orbit. For example, calculating the angular size of a lunar eclipse involves understanding the relationships between the Earth, Moon, and Sun, where tangents and secants help define the boundaries and angles of observation. This knowledge allows astronomers to accurately predict and interpret celestial events.
