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The biggest possible regular hexagon H is cut out of an equilateral triangle X. The biggest possible equilateral triangle Y is cut out from the hexagon H. What is the ratio of the areas of the equilateral triangles X and Y?
- a)5 : 1
- b)6 : 1
- c)8 : 1
- d)3 : 1
Correct answer is option 'D'. Can you explain this answer?
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Problem Analysis:
We need to find the ratio of the areas of the equilateral triangles X and Y, where the biggest possible regular hexagon H is cut out of an equilateral triangle X, and the biggest possible equilateral triangle Y is cut out from the hexagon H.
Given:
Let's assume the side of the equilateral triangle X be 'a'. Therefore,
- The perimeter of X = 3a
- The side of the regular hexagon H = a
- The perimeter of H = 6a
Calculation:
1. Area of X:
We know that the area of an equilateral triangle with side 'a' is given by:
Area of X = (√3/4) x a²
2. Area of H:
We know that the area of a regular hexagon with side 'a' is given by:
Area of H = (3√3/2) x a²
3. Side of the equilateral triangle Y:
The side of the equilateral triangle Y is equal to the length of the side of the hexagon H, which is 'a'.
4. Area of Y:
We know that the area of an equilateral triangle with side 'a' is given by:
Area of Y = (√3/4) x a²
5. Ratio of areas of X and Y:
Area of X/Area of Y = [ (√3/4) x a² ] / [ (√3/4) x a² ]
= 1:1
Therefore, the ratio of the areas of the equilateral triangles X and Y is 1:1.
Answer:
The correct answer is option 'D'.
We need to find the ratio of the areas of the equilateral triangles X and Y, where the biggest possible regular hexagon H is cut out of an equilateral triangle X, and the biggest possible equilateral triangle Y is cut out from the hexagon H.
Given:
Let's assume the side of the equilateral triangle X be 'a'. Therefore,
- The perimeter of X = 3a
- The side of the regular hexagon H = a
- The perimeter of H = 6a
Calculation:
1. Area of X:
We know that the area of an equilateral triangle with side 'a' is given by:
Area of X = (√3/4) x a²
2. Area of H:
We know that the area of a regular hexagon with side 'a' is given by:
Area of H = (3√3/2) x a²
3. Side of the equilateral triangle Y:
The side of the equilateral triangle Y is equal to the length of the side of the hexagon H, which is 'a'.
4. Area of Y:
We know that the area of an equilateral triangle with side 'a' is given by:
Area of Y = (√3/4) x a²
5. Ratio of areas of X and Y:
Area of X/Area of Y = [ (√3/4) x a² ] / [ (√3/4) x a² ]
= 1:1
Therefore, the ratio of the areas of the equilateral triangles X and Y is 1:1.
Answer:
The correct answer is option 'D'.
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