Rectangle R is formed by joining the centers of two congruent tangent circles and then drawing radii perpendicular to a common external tangent. If the perimeter of R is 60, what is its area?
A=30w-w^2
To find the area of the rectangle, we need to first determine its dimensions.
Let's assume that the radius of each of the congruent tangent circles is 'r'. Since the circles are congruent, they have the same radius.
The length of the rectangle R can be found by adding the diameter of both circles together. The diameter of a circle is twice the radius, so the length of the rectangle is 2r + 2r = 4r.
The width of the rectangle is the distance between the centers of the two circles. Since the circles are tangent to each other, the distance between their centers is 2r.
Thus, the dimensions of the rectangle are 4r (length) and 2r (width).
The perimeter of a rectangle is given by the formula:
P = 2(length + width)
We know that the perimeter of R is given as 60. Substituting the values we found earlier, we have:
60 = 2(4r + 2r)
Simplifying the equation, we have:
60 = 2(6r)
30 = 6r
r = 5
Now that we know the value of 'r', we can find the area of the rectangle:
Area of a rectangle = length × width
Substituting the values we found:
Area = (4r) × (2r)
Area = 4 × 5 × 2 × 5
Area = 40 × 10
Area = 400
Therefore, the area of the rectangle is 400 square units.