There are some parallelograms whose length of two sides is 2 cm and 1 cm, and one inside angle is 60 degrees.
These are connected at the vertex so that all of the 2 cm sides are parallel as shown in Fig.1 and Fig.2.
(1) A and B were connected as shown in Fig.1.
Find the area ratio of the sum of the area of two triangles painted black and the area of one parallelogram.
(2) C and D were connected as shown in Fig.2.
Find the area ratio of the sum of the area of four triangles painted black and the area of one parallelogram.
Solution
（1）
In Fig.1 the portion divided up and down by line AB is the same form and two triangles
painted black are congruent.
△ACB is created by drawing an auxiliary line as shown in Fig.3.
A red triangle turns into an equilateral triangle whose one side is 1 cm from the conditions in question.
The length of CB is 2cm + 1cm + 2cm + 2cm = 7cm.
△ACB and △ADG are homothetic and a homothetic ratio of AC : AD = 3 : 1.
Therefore, DG = CB × 1/3 = 7 × 1/3 = 7/3cm.
Since DE = 2cm, EG = 7/3 - 2 = 1/3cm.
△AHF and △EGH are homothetic and a homothetic ratio is AF : EG = 2 : 1/3 = 6 : 1.
An area ratio is 6 × 6 : 1 × 1 = 36 : 1.
The area ratio of △AHF of △EGH is 1 : 36.
In Fig.4 the area of △AEF is 1/2 times that of the parallelogram and △AHF is the 6 / (6+1) = 6/7 times of △AEF.
△AHF is the 1/2 × 6/7 = 3/7 times of the parallelogram.
Therefore, △EGH is the 3/7 × 1/36 = 1/84 time of the parallelogram.
The area ratio of a black area and the parallelogram is 1/84 × 2 : 1 = 1 : 42.
(2)
In Fig.5 ED = 2 + 1 + 2 + 2 + 1 + 2 + 2 = 12cm.
△CED and △CFI are homothetic and a homothetic ratio is 5 : 1.
FI = ED × 1/5 = 12 × 1/5 = 12/5cm.
Thus, GI = 12/5 - 2 = 2/5cm.
A homothetic ratio of △CJH and △JGI are 2/5 : 2 = 1 : 5 and an area ratio is 1 : 25.
△CJH is 1/2 × 5 / (1+5) = 5/12 of the area of parallelogram.
△JGI is 5/12 × 1/25 = 1/60 of the area of parallelogram. ----<1>
Next, since the homothetic ratio of △CED and △CKL is 5 : 2, KL = 12 × 2/5 = 24/5cm.
Thus, LN = KN - KN = (2 + 1 + 2) - 24/5 = 1/5cm.
△JGI and △LMN are homothetic and a homothetic ratio is GI : LN = 2/5 : 1/5 = 2 : 1 and an area ratio is 4 : 1.
According to <1>, the area of △LMN 1/60 × 1/4 = 1/240 of the area of a parallelogram----<2>
The area of a black triangle is △JGI × 2 + △LMN × 2.
<1> × 2 + <2> × 2 = 1/60 × 2 + 1/240 × 2 = 1/24 time.
The area ratio of the sum of black triangles and a parallelogram is 1/24 :1 = 1: 24.
These are connected at the vertex so that all of the 2 cm sides are parallel as shown in Fig.1 and Fig.2.
(1) A and B were connected as shown in Fig.1.
Find the area ratio of the sum of the area of two triangles painted black and the area of one parallelogram.
(2) C and D were connected as shown in Fig.2.
Find the area ratio of the sum of the area of four triangles painted black and the area of one parallelogram.
Answer
(1) 1 : 42
(2) 1 : 24
(2) 1 : 24
（1）
In Fig.1 the portion divided up and down by line AB is the same form and two triangles
painted black are congruent.
△ACB is created by drawing an auxiliary line as shown in Fig.3.
A red triangle turns into an equilateral triangle whose one side is 1 cm from the conditions in question.
The length of CB is 2cm + 1cm + 2cm + 2cm = 7cm.
△ACB and △ADG are homothetic and a homothetic ratio of AC : AD = 3 : 1.
Therefore, DG = CB × 1/3 = 7 × 1/3 = 7/3cm.
Since DE = 2cm, EG = 7/3 - 2 = 1/3cm.
△AHF and △EGH are homothetic and a homothetic ratio is AF : EG = 2 : 1/3 = 6 : 1.
An area ratio is 6 × 6 : 1 × 1 = 36 : 1.
The area ratio of △AHF of △EGH is 1 : 36.
In Fig.4 the area of △AEF is 1/2 times that of the parallelogram and △AHF is the 6 / (6+1) = 6/7 times of △AEF.
△AHF is the 1/2 × 6/7 = 3/7 times of the parallelogram.
Therefore, △EGH is the 3/7 × 1/36 = 1/84 time of the parallelogram.
The area ratio of a black area and the parallelogram is 1/84 × 2 : 1 = 1 : 42.
In Fig.5 ED = 2 + 1 + 2 + 2 + 1 + 2 + 2 = 12cm.
△CED and △CFI are homothetic and a homothetic ratio is 5 : 1.
FI = ED × 1/5 = 12 × 1/5 = 12/5cm.
Thus, GI = 12/5 - 2 = 2/5cm.
A homothetic ratio of △CJH and △JGI are 2/5 : 2 = 1 : 5 and an area ratio is 1 : 25.
△CJH is 1/2 × 5 / (1+5) = 5/12 of the area of parallelogram.
△JGI is 5/12 × 1/25 = 1/60 of the area of parallelogram. ----<1>
Next, since the homothetic ratio of △CED and △CKL is 5 : 2, KL = 12 × 2/5 = 24/5cm.
Thus, LN = KN - KN = (2 + 1 + 2) - 24/5 = 1/5cm.
△JGI and △LMN are homothetic and a homothetic ratio is GI : LN = 2/5 : 1/5 = 2 : 1 and an area ratio is 4 : 1.
According to <1>, the area of △LMN 1/60 × 1/4 = 1/240 of the area of a parallelogram----<2>
The area of a black triangle is △JGI × 2 + △LMN × 2.
<1> × 2 + <2> × 2 = 1/60 × 2 + 1/240 × 2 = 1/24 time.
The area ratio of the sum of black triangles and a parallelogram is 1/24 :1 = 1: 24.
