Let ABCDEF be a regular hexagon. What is the ratio of the area of the triangle ACE to that of the hexagon ABCDEF?a)1/3b)1/2c)2/3d)5/6Correct answer is option 'B'. Can you explain this answer?

Question

Answer

To find the ratio of the area of triangle ACE to the area of hexagon ABCDEF, we need to compare the areas of the two shapes.

A regular hexagon has six congruent equilateral triangles as its internal angles, with each internal angle measuring 120 degrees.

Since triangle ACE is an equilateral triangle, it also has internal angles measuring 60 degrees.

Now, let's consider the side lengths. Let's say the side length of the hexagon is 's'. Since it's a regular hexagon, all six sides have the same length.

In triangle ACE, all three sides are also of length 's' because it's an equilateral triangle.

To find the area of an equilateral triangle, we can use the formula:

Area = (sqrt(3)/4) * (side length)^2.

So, the area of triangle ACE is:

Area of ACE = (sqrt(3)/4) * s^2.

To find the area of the hexagon, we can divide it into six equilateral triangles. Each of these triangles has an area of:

Area of each triangle = (sqrt(3)/4) * s^2.

Since there are six such triangles in the hexagon, the total area of the hexagon is:

Area of hexagon = 6 * Area of each triangle

= 6 * [(sqrt(3)/4) * s^2]

= (3sqrt(3)/2) * s^2.

Now, let's calculate the ratio of the area of triangle ACE to the area of hexagon ABCDEF:

Area of ACE / Area of hexagon = [(sqrt(3)/4) * s^2] / [(3sqrt(3)/2) * s^2]

= [(sqrt(3)/4) / (3sqrt(3)/2)] * (s^2 / s^2)

= (1/2) * 1

= 1/2.

Therefore, the ratio of the area of triangle ACE to the area of hexagon ABCDEF is 1/2. So, option B is the correct answer.

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