A rectangle of length 4cm and width 3cm has one vertex at the origin of a system of axes and another vertex on a quarter of a circle whose center is also at the origin. Solve for the area
To solve for the area, we first need to find the length of the side of the rectangle that lies on the circumference of the quarter circle.
Since the center of the quarter circle is at the origin, the radius of the quarter circle is the distance between the origin and the point where the rectangle intersects the quarter circle.
We can find this distance using the Pythagorean theorem. The length of the rectangle is 4 cm and the width is 3 cm. The right triangle formed by the origin, the point where the rectangle intersects the quarter circle, and the end of the rectangle has a hypotenuse of 4 cm (the length of the rectangle) and a height of 3 cm (the width of the rectangle).
By applying the Pythagorean theorem, we can find the length of the side of the rectangle that lies on the circumference of the quarter circle:
c^2 = a^2 + b^2
c^2 = 4^2 + 3^2
c^2 = 16 + 9
c^2 = 25
Taking the square root of both sides:
c = √25 = 5 cm
So, the length of the side of the rectangle that lies on the circumference of the quarter circle is 5 cm.
Now we can calculate the area of the rectangle by multiplying its length (4 cm) by its width (3 cm):
Area of rectangle = Length × Width = 4 cm × 3 cm = 12 cm^2
Therefore, the area of the rectangle is 12 square centimeters.