A figure below is the figure which overlap 2 sheets of square papers with same size.
(1) In case of the area ratio of P and Q is P : Q = 3 : 2,
(1) In case of the area ratio of P and Q is P : Q = 3 : 2,
find the ratio of the length of CE and DE.
(2) In case of the area ratio of P and Q is P : Q = 2 : 1,
(2) In case of the area ratio of P and Q is P : Q = 2 : 1,
find the length ratio of the circumference of P and Q.
(3) In case of the length ratio of the circumference of P and Q is 5 : 3,
(3) In case of the length ratio of the circumference of P and Q is 5 : 3,
find the area ratio of P and Q.
Answer
(1) 2 : 3
(2) 3 : 2
(3) 4 : 1
Solution
(1) Assuming the length of AB is 1, the area of Q = 1 × CE × 1/2 × 2 = CE.
CE = 1 × 1 × 2/(2+3) = 2/5 and DE = 1 - 2/5 = 3/5.
Therefore CE : DE = 2 : 3.
(2) In this case, CE = 1 × 1/(1+2) = 1/3.
Therefore the length ratio of circumference of P and Q is
Answer
(1) 2 : 3
(2) 3 : 2
(3) 4 : 1
Solution
(1) Assuming the length of AB is 1, the area of Q = 1 × CE × 1/2 × 2 = CE.
CE = 1 × 1 × 2/(2+3) = 2/5 and DE = 1 - 2/5 = 3/5.
Therefore CE : DE = 2 : 3.
(2) In this case, CE = 1 × 1/(1+2) = 1/3.
Therefore the length ratio of circumference of P and Q is
(1 × 4) : (1 × 2 + 1/3 × 2) = 4 : 8/3 = 3 : 2.
(3) When the length of the circumference of P is set to be 5, the length of one side of the square is to be 5/4.
In this case the length of CE = (3 - 5/4 × 2) × 1/2 = 1/4.
Thus the area of Q = 5/4 × 1/4 × 1/2 × 2 = 5/16.
The area of P = 5/4 × 5/4 - 5/16 = 20/16.
Therefore the area ratio of P and Q = 20/16 : 5 /16 = 4 : 1.
(3) When the length of the circumference of P is set to be 5, the length of one side of the square is to be 5/4.
In this case the length of CE = (3 - 5/4 × 2) × 1/2 = 1/4.
Thus the area of Q = 5/4 × 1/4 × 1/2 × 2 = 5/16.
The area of P = 5/4 × 5/4 - 5/16 = 20/16.
Therefore the area ratio of P and Q = 20/16 : 5 /16 = 4 : 1.
