The opening angle of the projector is 60, which is the closest distance from the projector to the screen.

The opening angle of the projector is 60, which is the closest distance from the projector to the screen. And 3√3 m high so that it is fully illuminated.

We represent the data under the condition of the value in the form △ ABC: ∠B = 90 °, ∠C = 60 ° – projector opening angle, AB = 3√3 (screen height A) and BC (distance from projector to screen A) – legs, AC – hypotenuse (since it lies opposite the right angle).

1. By the theorem on the sum of the angles of a triangle:

∠A + ∠B + ∠C = 180 °;

∠A + 90 ° + 60 ° = 180 °;

∠A + 150 ° = 180 °;

∠A = 180 ° – 150 °;

∠A = 30 °.

2. Since the leg BC lies opposite the angle equal to 30 °, it is equal to half of the hypotenuse AC:

BC = AC / 2.

3. By the Pythagorean theorem:

AC² = AB² + BC²;

AC² = (3√3) ² + (AC / 2) ²;

AC² = 9 * 3 + AC² / 4;

AC² – AC² / 4 = 27;

(4 * AC² – AC²) / 4 = 27;

(3 * AC²) / 4 = 27;

AC² = (4 * 27) / 3;

AC² = 4 * 9;

AC² = 36;

AC = √36;

AC = 6 m.

4. Find the length BC:

BC = AC / 2 = 6/2 = 3 m.

Answer: 3 meters.