Connected parallelograms /

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.

Answer

(1) 1 : 42

(2) 1 : 24

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 x 1/3 = 7 x 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 : ⅓ = 6 : 1.

An area ratio is 6 x 6 : 1 x 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 ½ x 6/7 = 3/7 times of the parallelogram.

Therefore, △EGH is the 3/7 x 1/36 = 1/84 time of the parallelogram.

The area ratio of a black area and the parallelogram is 1/84 x 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 x ⅕ = 12 x ⅕ = 12/5cm. Thus, GI = 12/5 - 2 = 2/5cm.

A homothetic ratio of △CJH and △JGI are ⅖ : 2 = 1 : 5 and an area ratio is 1 : 25.

△CJH is ½ x 5 / (1+5) = 5/12 of the area of parallelogram.

△JGI is 5/12 x 1/25 = 1/60 of the area of parallelogram. ----<1>

Next, since the homothetic ratio of △CED and △CKL is 5 : 2, KL = 12 x ⅖ = 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 = ⅖ : ⅕ = 2 : 1 and an area ratio is 4 : 1.

According to <1>, the area of △LMN 1/60 x ¼ = 1/240 of the area of a parallelogram----<2>

The area of a black triangle is △JGI x 2 + △LMN x 2.

<1> x 2 + <2> x 2 = 1/60 x 2 + 1/240 x 2 =1/24 time.

The area ratio of the sum of black triangles and a parallelogram is 1/24 :1 = 1: 24.