This is a continuation of Exercise 7. An equilateral...

This is a continuation of Exercise 7. An equilateral triangle is one in which all sides have the same length. Show that if each side has length $a$, then the area of the triangle is
$$
\frac{\sqrt{3} a^2}{4} .
$$

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Key Concepts

Altitude Determination

Determining the altitude (or height) of an equilateral triangle involves dropping a perpendicular from one vertex to the midpoint of the opposite side. This process divides the equilateral triangle into two right triangles, enabling the use of the Pythagorean theorem to derive the height, which is key for computing the area.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in mathematics that relates the lengths of the sides in a right-angled triangle. It is used to calculate the height of an equilateral triangle by splitting it into two congruent right triangles, thereby establishing the relationship between the side length and the altitude.

Triangle Area Formula

The area of any triangle can be found using the formula (1/2) × base × height. This basic formula is widely applied in geometry to relate one-dimensional measures (base and height) to the two-dimensional measure of area.

Equilateral Triangle

An equilateral triangle is a type of triangle where all three sides and all three internal angles are equal. This symmetry simplifies many calculations, such as determining the altitude and area, making it a fundamental example in geometric studies.

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