To find the area of the triangular region ABC, we start by analyzing the given information. We have a circle with a center O and a radius of 1. This means that any point on the circumference of the circle is 1 unit away from the center O.
The segment BC is also given to be 1 unit long. We can visualize this situation by placing point O at the origin of a coordinate system. Let’s say point B is located at (1, 0). Since the radius is 1, point C must also lie on the circumference of the circle, and we can denote its coordinates as (x, y) such that:
x² + y² = 1 (circumference equation)
Since BC has a length of 1, we can use the distance formula to find out where point C can be located. The distance between points B (1, 0) and C (x, y) can be expressed as:
Distance BC = √[(x – 1)² + (y – 0)²] = 1
Squaring both sides gives:
(x – 1)² + y² = 1
Now, substituting y² from the first equation into this equation yields:
(x – 1)² + (1 – x²) = 1
Expanding this, we have:
(x² – 2x + 1) + 1 – x² = 1
-2x + 2 = 1
Solving for x gives:
−2x = -1
x = 0.5
Using x = 0.5 to find y, substitute back into the circle equation:
(0.5)² + y² = 1
0.25 + y² = 1
y² = 0.75
y = ±√0.75 = ±(√3/2)
So point C can be either (0.5, √3/2) or (0.5, -√3/2).
To find the area of triangle ABC, we can use the formula:
Area = 1/2 * base * height
Taking BC as the base (length = 1) and the height from point A (which we can consider to be the center O, (0, 0)) along the perpendicular dropped from A to line BC:
Since points B and C are both 1 unit from the center O, the height of triangle ABC can be determined by the vertical distance from O to line BC, which can be derived from the coordinates we’ve found. The height corresponds to the y-coordinate of points C, which can take the value of √3/2.
Consequently, the area of triangle ABC is:
Area = 1/2 * base * height = 1/2 * 1 * (√3/2) = √3/4.
Therefore, the area of the triangular region ABC is √3/4 square units.