By putting the equilateral triangles with 1cm one side in order without any gap nor overlap, a big equilateral triangle is made.
(1) How many equilateral triangles with 1cm one side are used for the equilateral triangle with 4cm one side?
(2) The equilateral triangle of 1cm, 2cm, 3cm, 4cm ---- one side is named as the 1st, the 2nd, the 3rd, the 4th, ----- equilateral triangle, respectively.
Answer the following questions.
① How many equilateral triangles with 1cm one side are used for the 26th equilateral triangle?
② I made two equilateral triangles, one is a certain ordinal number and another is the following.
There are in total 1013 pieces of equilateral triangles with 1cm one side are used for the two equilateral triangles.
Find the ordinal number of each equilateral triangle.
(1) 16 pieces
(2) ① 676 pieces
② 22nd and 23rd
Solution
(1) The equilateral triangle with 4cm one side and the equilateral triangle with 1cm one side are homothetic and a homothetic ratio is 4 : 1.
An area ratio is 4 × 4 : 1 × 1 = 16 : 1.
The number of the equilateral triangles of the area 1 required to make the equilateral triangle of the area 16 is 16 / 1 = 16 piece.
(2) ① Since the length of one side of the 26th equilateral triangle is 26cm, the homothetic ratio with the equilateral triangle 1cm one side is 16 : 1.
Considering the same way as (1), the number of the equilateral triangles with 1cm one side is 26 × 26 = 676 pieces.
② Since 1013 / 2 = 506.5, it is found that the number which makes a certain ordinal number of equilateral triangle is 506 or less, and the following ordinal number is 506 or more.
Since the number which makes an equilateral triangle is a square number, examine the square number near 506.
According to the calculation as 21 × 21 = 441, 22 × 22 = 484, 23 × 23 = 529, and 24 × 24 = 576, it turns out that 484 + 529 =1013 is applied to conditions.
Therefore, these two triangles are the 22nd and the 23rd.
(1) How many equilateral triangles with 1cm one side are used for the equilateral triangle with 4cm one side?
(2) The equilateral triangle of 1cm, 2cm, 3cm, 4cm ---- one side is named as the 1st, the 2nd, the 3rd, the 4th, ----- equilateral triangle, respectively.
Answer the following questions.
① How many equilateral triangles with 1cm one side are used for the 26th equilateral triangle?
② I made two equilateral triangles, one is a certain ordinal number and another is the following.
There are in total 1013 pieces of equilateral triangles with 1cm one side are used for the two equilateral triangles.
Find the ordinal number of each equilateral triangle.
(2) ① 676 pieces
② 22nd and 23rd
Solution
(1) The equilateral triangle with 4cm one side and the equilateral triangle with 1cm one side are homothetic and a homothetic ratio is 4 : 1.
An area ratio is 4 × 4 : 1 × 1 = 16 : 1.
The number of the equilateral triangles of the area 1 required to make the equilateral triangle of the area 16 is 16 / 1 = 16 piece.
(2) ① Since the length of one side of the 26th equilateral triangle is 26cm, the homothetic ratio with the equilateral triangle 1cm one side is 16 : 1.
Considering the same way as (1), the number of the equilateral triangles with 1cm one side is 26 × 26 = 676 pieces.
② Since 1013 / 2 = 506.5, it is found that the number which makes a certain ordinal number of equilateral triangle is 506 or less, and the following ordinal number is 506 or more.
Since the number which makes an equilateral triangle is a square number, examine the square number near 506.
According to the calculation as 21 × 21 = 441, 22 × 22 = 484, 23 × 23 = 529, and 24 × 24 = 576, it turns out that 484 + 529 =1013 is applied to conditions.
Therefore, these two triangles are the 22nd and the 23rd.
