A rectangular pentagonal garden plot has centre of symmetry O and an area of 50m^2. Find the distance OA.
What is a rectangular pentagonal garden?
Where is point A ?
To find the distance OA, we need to know the dimensions of the rectangular pentagonal garden plot.
Let's assume the length of the rectangle is L and the width is W.
Since the garden plot has a center of symmetry, we can divide it into two equal halves. One half of the garden plot is a rectangle with dimensions L/2 and W, and the other half is a pentagon.
The area of the rectangle is given by:
Area of rectangle = (L/2) * W
Given that the total area of the garden plot is 50m^2, we have:
(L/2) * W + Area of pentagon = 50
We know that the area of a pentagon can be calculated using the formula:
Area of pentagon = (5/4) * s^2 * cot(π/5)
where s is the side length of the pentagon.
Now, let's find the side length of the pentagon. Since the rectangle is symmetrical, the distance between the center of symmetry and one of the vertices of the pentagon will be the same as the width of the rectangle.
So, OA = W.
We can now substitute the formulas and variables into the equation:
(L/2) * W + (5/4) * s^2 * cot(π/5) = 50
Since we do not have enough information to directly calculate the dimensions of the rectangle or the side length of the pentagon, we cannot determine the exact value of OA without additional information.
To find the distance OA, we need to determine the dimensions of the rectangular pentagonal garden plot.
A rectangular pentagon is a polygon with five angles, four of which are right angles. Since the garden plot is rectangular, we know that all four angles are right angles.
Let's assume the length of the garden plot is x meters and the width is y meters.
Given that the area of the garden plot is 50m^2, we have the equation:
xy = 50
Since the garden plot has a center of symmetry O, the distance OA will be half the length of the plot. Therefore, we need to find the value of x/2.
To solve the equation, we can isolate one variable and substitute it into the other equation. Let's solve for x:
From xy = 50, we can rearrange the equation to solve for x:
x = 50/y
Now, substitute this value into the equation for the area:
(50/y) * y = 50
Simplifying the equation, we have:
50 = 50
This equation is always true, so it doesn't provide us with any valuable information.
To find the value of x/2 (distance OA), we need more information.
I think we can safely assume that it's a "regular" pentagonal garden, and that OA is the apothem. If so, then the 5 isosceles triangles have a vertex angle of 72°
If we call the side length 2s and the apothem length a, then we have
s/a = tan36°
That makes the area of each isosceles triangle s*a = a^2 tan36°
The area of the pentagon is thus 5a^2 tan36°
Now we have
5a^2 tan36° = 50
OA = a = √(10 cot36°)