As shown in a figure, the circle A centering on the point A with a radius of 3 cm and the circle B centering on the point B with a radius of 2 cm have touched.
The points C and D are points on the circumference of the circle B.
The points E and F are points on the circumference of the circle A.
Three point A, E and C are on a straight line and three point A, F and D are also on a straight line.
Find the area ratio of a quadrangle ACBD and the triangle AEF.
The points C and D are points on the circumference of the circle B.
The points E and F are points on the circumference of the circle A.
Three point A, E and C are on a straight line and three point A, F and D are also on a straight line.
Find the area ratio of a quadrangle ACBD and the triangle AEF.
Answer
△AFG and △ABD are homothetic and a homothetic ratio is AF : AB = 3 : 5.
An area ratio is 3 × 3 : 5 × 5 = 9 : 25.
The area of △AEF is twice of △AFG and the area of a quadrangle ACBD is also the twice of △ABD.
The area ratio of △AEF and a quadrangle ACBD is also 9 : 25.
25 : 9
Solution
Since the length of AB is equal to the sum length of each radius of the circle A and the circle B, it is 3 + 2 = 5cm.
△AFG and △ABD are homothetic and a homothetic ratio is AF : AB = 3 : 5.
An area ratio is 3 × 3 : 5 × 5 = 9 : 25.
The area of △AEF is twice of △AFG and the area of a quadrangle ACBD is also the twice of △ABD.
The area ratio of △AEF and a quadrangle ACBD is also 9 : 25.
