Provide geometric results that you will need in working...

Provide geometric results that you will need in working Exercises $51-54 .$ For Exercise $50,$ you need to know that a segment of a circle is the region bounded by an arc of the circle and its chord. In the accompanying figure, the red region is a segment. (The white region also is a segment.) (FIGURE CAN'T COPY) In the accompanying figure, $\triangle A B C$ is equilateral and $s$ denotes the length of each side. The arc in the figure is a portion of a circle with center $A$ and radius $A B=s .$ Use the result in Exercise 49 and a formula from this section to show that the area of the shaded segment in the figure is given by $$ s^{2}\left(\frac{2 \pi-3 \sqrt{3}}{12}\right) $$

Provide geometric results that you will need in working Exercises $51-54 .$ For Exercise $50,$ you need to know that a segment of a circle is the region bounded by an arc of the circle and its chord. In the accompanying figure, the red region is a segment. (The white region also is a segment.)
(FIGURE CAN'T COPY)
In the accompanying figure, $\triangle A B C$ is equilateral and $s$ denotes the length of each side. The arc in the figure is a portion of a circle with center $A$ and radius $A B=s .$ Use the result in Exercise 49 and a formula from this section to show that the area of the shaded segment in the figure is given by
$$
s^{2}\left(\frac{2 \pi-3 \sqrt{3}}{12}\right)
$$

Key Concepts

Composite Area Techniques

Composite area techniques involve finding the area of a complex shape by decomposing it into simpler regions. For example, calculating the area of a segment of a circle involves computing the area of a sector and subtracting the area of an inscribed triangle. This approach is broadly applicable in various geometric problems involving overlapping or adjacent shapes.

Triangle Area Calculation

Triangles often appear in problems involving circles, especially when chords and radii form a triangle. Knowing how to calculate the area of a triangle, whether through standard formulas or specialized ones (as for equilateral triangles), is important for subtracting the triangular area from the sector area to determine the area of a segment.

Central Angle and Arc

The central angle is the angle with its vertex at the center of a circle, and its sides intersect the circle, defining the arc. This concept is crucial because both the area of the circular sector and the length of the arc depend directly on this angle, forming a bridge between angular and linear measurements in circle geometry.

Circular Sector

A circular sector is a portion of a circle enclosed by two radii and the arc between them. Its area is proportional to the measure of the central angle that the radii form. The area formula, (?/2?)?r² or equivalently ½r²? (with ? in radians), is essential in computations involving segments and portions of a circle.

Circle Segment

A circle segment is the region of a circle bounded by an arc and the chord connecting the arc’s endpoints. The area of a circular segment is typically calculated by subtracting the area of the triangle formed by the chord and the two radii from the area of the corresponding circular sector.

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