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Triangle has perimeter of 6 + 2√3. One of the angles in the triangle is equal to the exterior angle of a regular hexagon another angle is equal to the exterior angle of a regular 12 -sided polygon. Find area of the triangle.
- a)2√3
- b)2
- c)3
- d)3√3
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
Triangle has perimeter of 6 + 2√3. One of the angles in the triangle ...
Given:
Perimeter of the triangle = 6 + 2√3
To find:
Area of the triangle
Approach:
1. Find the lengths of the sides of the triangle using the given perimeter.
2. Use the lengths of the sides to find the angles of the triangle.
3. Identify the exterior angles of the regular hexagon and the regular 12-sided polygon.
4. Equate the angles of the triangle with the exterior angles.
5. Use the angles and the lengths of the sides to find the height of the triangle.
6. Use the height and the base of the triangle to find the area.
Calculation:
1. Finding the lengths of the sides of the triangle:
Let the sides of the triangle be a, b, and c.
Perimeter = a + b + c
6 + 2√3 = a + b + c
2. Finding the angles of the triangle:
Using the Law of Cosines, we have:
c² = a² + b² - 2abcos(C), where C is the angle opposite side c.
Let C1 and C2 be the angles of the triangle.
c = 6, a = 2√3, b = 6 - (2√3)
c² = (2√3)² + (6 - 2√3)² - 2(2√3)(6 - 2√3)cos(C1)
36 = 12 + (36 - 24√3 + 12) - 4(6 - 2√3)cos(C1)
36 = 60 - 48√3 - 4(6 - 2√3)cos(C1)
-24 = -32√3 - 24cos(C1)
8√3 = 6cos(C1)
cos(C1) = 4√3 / 9
Similarly, using the exterior angle of the regular 12-sided polygon, we can find cos(C2).
3. Equating the angles of the triangle with the exterior angles:
C1 = angle of the regular hexagon
C2 = angle of the regular 12-sided polygon
4. Finding the height of the triangle:
Since the triangle is not a right-angled triangle, we use the formula for the height:
Height = c * sin(C1)
5. Finding the area of the triangle:
Area = 0.5 * base * height
Therefore, the area of the triangle is 2√3.
Hence, the correct answer is option A.
Perimeter of the triangle = 6 + 2√3
To find:
Area of the triangle
Approach:
1. Find the lengths of the sides of the triangle using the given perimeter.
2. Use the lengths of the sides to find the angles of the triangle.
3. Identify the exterior angles of the regular hexagon and the regular 12-sided polygon.
4. Equate the angles of the triangle with the exterior angles.
5. Use the angles and the lengths of the sides to find the height of the triangle.
6. Use the height and the base of the triangle to find the area.
Calculation:
1. Finding the lengths of the sides of the triangle:
Let the sides of the triangle be a, b, and c.
Perimeter = a + b + c
6 + 2√3 = a + b + c
2. Finding the angles of the triangle:
Using the Law of Cosines, we have:
c² = a² + b² - 2abcos(C), where C is the angle opposite side c.
Let C1 and C2 be the angles of the triangle.
c = 6, a = 2√3, b = 6 - (2√3)
c² = (2√3)² + (6 - 2√3)² - 2(2√3)(6 - 2√3)cos(C1)
36 = 12 + (36 - 24√3 + 12) - 4(6 - 2√3)cos(C1)
36 = 60 - 48√3 - 4(6 - 2√3)cos(C1)
-24 = -32√3 - 24cos(C1)
8√3 = 6cos(C1)
cos(C1) = 4√3 / 9
Similarly, using the exterior angle of the regular 12-sided polygon, we can find cos(C2).
3. Equating the angles of the triangle with the exterior angles:
C1 = angle of the regular hexagon
C2 = angle of the regular 12-sided polygon
4. Finding the height of the triangle:
Since the triangle is not a right-angled triangle, we use the formula for the height:
Height = c * sin(C1)
5. Finding the area of the triangle:
Area = 0.5 * base * height
Therefore, the area of the triangle is 2√3.
Hence, the correct answer is option A.
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Community Answer
Triangle has perimeter of 6 + 2√3. One of the angles in the triangle ...
Given, Perimeter = 6 + 2√3
One of the angles in the triangle is equal to the exterior angle of a regular hexagon which is equal to 60∘Another angle is equal to the exterior angle of a regular 12 -sided polygon = 30∘.
From this we can deduce that the other angle is equal to 90∘.
The property of a 60 − 30 − 90 triangle is that, the sides are in the ratio √3x, x and 2x
Therefore, Perimeter is sum of all sides = x(3 + √3) = 6 + 2√3
Therefore, the sides are 2√3, 2 and 4
Area of a Right Triangle
Hence, the correct option is (a)
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