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) Show that the area of an equilateral triangle of side $s$ is given by $\frac{\sqrt{3}}{4}$ Hint: Draw an altitude and use the Pythagorean theorem. (This exercise does not require any knowledge of radian measure. $)$

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)
Show that the area of an equilateral triangle of side $s$ is given by
$\frac{\sqrt{3}}{4}$
Hint: Draw an altitude and use the Pythagorean theorem. (This exercise does not require any knowledge of radian measure. $)$

Key Concepts

Circular Segment

A circular segment is defined as the region of a circle that is bounded by an arc and its corresponding chord. Recognizing and understanding the properties of circular segments is important for solving geometric exercises that involve portions of circles or relate circular areas with linear segments.

Area of a Triangle

The area of any triangle can be calculated using the formula one-half the product of its base and corresponding height. For an equilateral triangle, once the altitude is determined via the Pythagorean theorem, this formula is used to derive the area expression in terms of the side length.

Pythagorean Theorem

The Pythagorean theorem applies to right triangles and states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This theorem is instrumental in calculating the altitude of an equilateral triangle by relating the side lengths of the triangle's split right-angled components.

Altitude

The altitude of a triangle is a perpendicular segment dropped from a vertex to the line containing the opposite side. In an equilateral triangle, the altitude not only provides a means to calculate the triangle’s area but also splits the triangle into two congruent right-angled triangles, thereby simplifying calculations using the Pythagorean theorem.

Equilateral Triangle

An equilateral triangle is a triangle in which all three sides are of equal length and all three angles are congruent, each measuring 60°. This symmetry leads to many useful properties including congruent altitudes, medians, and angle bisectors, which facilitate calculations such as finding the area or determining other geometric relationships.

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