Question

In the construction of an equilateral triangle with both inscribed and circumscribed circles, it can be seen that the ratio of the area of the inscribed circle to the area of the circumscribed circle is 1:2 .

A.True
B.False

Answer 1

The ratio of the area of the inscribed circle to the area of the circumscribed circle in an equilateral triangle is not 1:2, as this is based on the relationship between the side length of the triangle and the radii of the circles, which does not provide this ratio.

Step-by-step explanation:

The statement that the ratio of the area of the inscribed circle to the area of the circumscribed circle in an equilateral triangle is 1:2 is False. To understand why, we can use the formula for the area of a circle, which is πr², where r is the radius of the circle, and the known relationship in an equilateral triangle between the side length and the radii of the inscribed and circumscribed circles.

For an equilateral triangle with side length a and an inscribed circle (incircle) with radius r, the area of the incircle is πr². The radius of the circumcircle (circumscribed circle) is longer and can be found using the formula R = a/√3. The area of the circumcircle would then be πR². The ratio of the areas is dependent on the squares of the radii, which do not fit into the ratio of 1:2.