There are circle with radius 2cm centered on point O and square ABCD whose length of one side is 6cm as shown in the figure.
Point P moves inside and on the circumference of the circle.
(1) When point Q moves on the circumference of the square, find the length of the circumference of the figure which is made by the middle point of the line segment OQ moves.
(2) Find the area of the figure made by the middle point of the line segment AP moves.
(3) When point Q moves on the circumference of the square, find the area of the figure made by the middle point of the line segment PQ moves.
Answer
(1) 12 cm
(2) 3.14 cm2
(3) 23.14 cm2
Solution
(1) The figure made when the middle point of the line segment OQ moves is shown as a red square in the figure below.
The length of one side of this square is 1/2 of the length of one of square ABCD.
Therefore 6 cm × 1/2 × 4 = 12 cm.
(2) The figure made when the middle point of the line segment AP moves is shown as a red circle in the figure below.
The radius of this red circle is 1/2 of the radius of the circle O and it is 2 cm × 1/2 = 1 cm.
Therefore the area is 1 × 1 × 3.14 = 3.14 cm2.
(3) Considering the result of (1) and (2) as a reference, the figure made when the middle point of the line segment PQ moves is shown as a red shadow in the figure below.
The length of one side of outside red square is 1 × 2 × 2 + 1 = 5 cm.
Therefore the area is 5 × 5 - 1 × 1 - (2 × 2 - 1 × 1 × 3.14) = 23.14 cm2.
