From point F outside the circle, two secants are drawn. The first secant intersects the circle at points A and B (A between F and B), and the second secant intersects the circle at points C and D (C between F and D). Chords AD and BC intersect at point E. Find the degree measure of angle BED if ∠BFD = 36° and arc AC : arc BD = 2 : 5.

Analysis of the problem:

We are given a diagram with a circle and two secants drawn from an external point F. The secants intersect the circle at points A, B and C, D respectively. The chords AD and BC intersect inside the circle at point E. We are given the angle ∠BFD = 36° and the ratio of the measures of arcs AC and BD as 2:5.

We need to find the measure of angle BED.

Key Geometric Properties:

• The angle formed by two secants drawn from an external point to a circle is half the difference of the measures of the intercepted arcs. So, ∠BFD = 1/2 * (arc BD - arc AC).
• The angle formed by two intersecting chords inside a circle is half the sum of the measures of the intercepted arcs. So, ∠BED = 1/2 * (arc BD + arc AC).

Solution:

1. Using the external secant angle formula:
• We are given ∠BFD = 36°.
• Let the measure of arc AC be 2x and the measure of arc BD be 5x, based on the given ratio.
• According to the formula for the angle formed by two secants:

∠BFD = 1/2 * (m(arc BD) - m(arc AC))

36° = 1/2 * (5x - 2x)

36° = 1/2 * (3x)

72° = 3x

x = 72° / 3

x = 24°

2. Calculate the measures of the arcs:
• Measure of arc AC = 2x = 2 * 24° = 48°.
• Measure of arc BD = 5x = 5 * 24° = 120°.
3. Using the intersecting chords angle formula:
• Now we need to find ∠BED. The chords AD and BC intersect at E.
• According to the formula for the angle formed by two intersecting chords:

∠BED = 1/2 * (m(arc BD) + m(arc AC))

∠BED = 1/2 * (120° + 48°)

∠BED = 1/2 * (168°)

∠BED = 84°

Final Answer:

Ответ: 84