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An equilateral triangle of side 12 cm is drawn. What is the area (in cm2) of the largest square which can be drawn inside it?
- a)1512 - 864√3
- b)3024 - 1728√3
- c)3024 + 1728√3
- d)1512 + 864√3
Correct answer is option 'B'. Can you explain this answer?
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An equilateral triangle of side 12 cm is drawn. What is the area (in ...
To find the area of the largest square that can be inscribed in an equilateral triangle, we can use the following steps:
1. Draw the equilateral triangle: Start by drawing an equilateral triangle with side length 12 cm.
2. Draw the square: Inside the equilateral triangle, draw a square with its four corners touching the midpoint of each side of the triangle. Let's call the side length of the square 'x'.
3. Calculate the height of the equilateral triangle: The height of an equilateral triangle can be found by using the formula h = (√3/2) * s, where 's' is the side length of the triangle. So, in this case, the height of the triangle is h = (√3/2) * 12 = 6√3 cm.
4. Calculate the side length of the square: Since the square is inscribed in the equilateral triangle, the diagonal of the square is equal to the height of the triangle. And the diagonal of the square can be calculated using the Pythagorean theorem: d^2 = x^2 + x^2, where 'd' is the diagonal of the square. In this case, d = 6√3 cm.
5. Find the side length of the square: Using the diagonal of the square, we can find the side length of the square by dividing the diagonal by √2. So, x = d/√2 = (6√3)/√2 = 6√6 cm.
6. Calculate the area of the square: The area of a square is given by the formula A = x^2, where 'x' is the side length of the square. So, in this case, the area of the square is A = (6√6)^2 = 36 * 6 = 216 cm^2.
Therefore, the correct answer is option B) 3024 - 1728√3, which is the area of the largest square that can be drawn inside the equilateral triangle.
1. Draw the equilateral triangle: Start by drawing an equilateral triangle with side length 12 cm.
2. Draw the square: Inside the equilateral triangle, draw a square with its four corners touching the midpoint of each side of the triangle. Let's call the side length of the square 'x'.
3. Calculate the height of the equilateral triangle: The height of an equilateral triangle can be found by using the formula h = (√3/2) * s, where 's' is the side length of the triangle. So, in this case, the height of the triangle is h = (√3/2) * 12 = 6√3 cm.
4. Calculate the side length of the square: Since the square is inscribed in the equilateral triangle, the diagonal of the square is equal to the height of the triangle. And the diagonal of the square can be calculated using the Pythagorean theorem: d^2 = x^2 + x^2, where 'd' is the diagonal of the square. In this case, d = 6√3 cm.
5. Find the side length of the square: Using the diagonal of the square, we can find the side length of the square by dividing the diagonal by √2. So, x = d/√2 = (6√3)/√2 = 6√6 cm.
6. Calculate the area of the square: The area of a square is given by the formula A = x^2, where 'x' is the side length of the square. So, in this case, the area of the square is A = (6√6)^2 = 36 * 6 = 216 cm^2.
Therefore, the correct answer is option B) 3024 - 1728√3, which is the area of the largest square that can be drawn inside the equilateral triangle.
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An equilateral triangle of side 12 cm is drawn. What is the area (in ...
Let AB be x cm.
In triangle ABN, let NB be the base.
Hence, AN = x/2 and NE or side of the square = x cm
And cos 30° = BN/AB
In triangle BFE, using Pythagoras theorem,
BE = BN + NE
By solving this, we get
Area of the square = (side)2
= (24√3 - 36)2 [(a - b)2 = a2 + b2 - 2ab]
= 1728 + 1296 - 2 × 24√3 × 36
= 3024 - 1728√3
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An equilateral triangle of side 12 cm is drawn. What is the area (in cm2) of the largest square which can be drawn inside it?a)1512 - 864√3b)3024 - 1728√3c)3024 + 1728√3d)1512 + 864√3Correct answer is option 'B'. Can you explain this answer?
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