The area of an equilateral triangle is 3√3 cm2. Find the radius of a circle inscribed in a triangle.
July 27, 2021 | education|
Let us denote by a the length of the side of this equilateral triangle.
According to the condition of the problem, the area of this triangle is 3√3 cm ^ 2.
Since each angle of an equilateral triangle is 60 °, applying the formula for the area of a triangle along two sides and the angle between them, we can compose the following equation:
a * a * sin (60 °) / 2 = 3√3,
solving which, we get:
a ^ 2 * √3 / 2 = 3√3;
a ^ 2 = 3√3 / (√3 / 2);
a ^ 2 = 6;
a = √6 cm.
Applying the formula for the area of a triangle in terms of the radius r of the inscribed circle, we find r:
r = 2 * 3√3 / (√6 + √6 + √6) = 2 * 3√3 / (3√6) = 2 / √2 = √2 cm.
Answer: the radius of the inscribed circle is √2 cm.
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