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In the given figure, ABCDEF is a regular hexagon of side 12 cm. P, Q and R are the mid points of the sides AB, CD and EF respectively. What is the area (in cm²) of triangle PQR ?
- a)27√6
- b)81√3
- c)54√3
- d)54√6
Correct answer is option 'B'. Can you explain this answer?
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In the given figure, ABCDEF is a regular hexagon of side 12 cm. P, Q a...
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In the given figure, ABCDEF is a regular hexagon of side 12 cm. P, Q a...
To find the area of the shaded region, we need to find the area of the regular hexagon ABCDEF and subtract the areas of the three triangles formed by connecting the midpoints of the sides.
The area of a regular hexagon can be found using the formula:
Area = (3√3/2) * (side length)^2
Plugging in the values, we get:
Area of hexagon = (3√3/2) * (12 cm)^2
= (3√3/2) * 144 cm^2
= 216√3 cm^2
Now, let's find the area of the triangles.
Triangle APB:
The base of this triangle is half the side length of the hexagon, which is 6 cm.
The height of this triangle can be found using the Pythagorean theorem:
Height^2 = (side length)^2 - (base/2)^2
Height^2 = 144 cm^2 - 9 cm^2
Height^2 = 135 cm^2
Height = √135 cm
Area of triangle APB = (1/2) * base * height
= (1/2) * 6 cm * √135 cm
= 3√135 cm^2
Similarly, we can find the areas of triangles CQD and FRE.
Area of triangle CQD = 3√135 cm^2
Area of triangle FRE = 3√135 cm^2
Now, let's find the area of the shaded region by subtracting the areas of the triangles from the area of the hexagon:
Area of shaded region = Area of hexagon - Area of triangle APB - Area of triangle CQD - Area of triangle FRE
= 216√3 cm^2 - 3√135 cm^2 - 3√135 cm^2 - 3√135 cm^2
= 216√3 cm^2 - 9√135 cm^2
= (216 - 9)√3 cm^2
= 207√3 cm^2
Therefore, the area of the shaded region is 207√3 cm^2.
The area of a regular hexagon can be found using the formula:
Area = (3√3/2) * (side length)^2
Plugging in the values, we get:
Area of hexagon = (3√3/2) * (12 cm)^2
= (3√3/2) * 144 cm^2
= 216√3 cm^2
Now, let's find the area of the triangles.
Triangle APB:
The base of this triangle is half the side length of the hexagon, which is 6 cm.
The height of this triangle can be found using the Pythagorean theorem:
Height^2 = (side length)^2 - (base/2)^2
Height^2 = 144 cm^2 - 9 cm^2
Height^2 = 135 cm^2
Height = √135 cm
Area of triangle APB = (1/2) * base * height
= (1/2) * 6 cm * √135 cm
= 3√135 cm^2
Similarly, we can find the areas of triangles CQD and FRE.
Area of triangle CQD = 3√135 cm^2
Area of triangle FRE = 3√135 cm^2
Now, let's find the area of the shaded region by subtracting the areas of the triangles from the area of the hexagon:
Area of shaded region = Area of hexagon - Area of triangle APB - Area of triangle CQD - Area of triangle FRE
= 216√3 cm^2 - 3√135 cm^2 - 3√135 cm^2 - 3√135 cm^2
= 216√3 cm^2 - 9√135 cm^2
= (216 - 9)√3 cm^2
= 207√3 cm^2
Therefore, the area of the shaded region is 207√3 cm^2.
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