Question

1. finding a pattern in the pattern shown, each small triangle is an equilateral triangle with an area of 1 square unit.

a. explain how you know that any triangle made out of equilateral triangles is equilateral.
b. find the areas of the first four triangles in the pattern.
c. describe any patterns in the areas. predict the area of the seventh triangle in the pattern. explain your reasoning.

triangle	area
!triangle made of 4 small equilateral triangles
!triangle made of 9 small equilateral triangles
!triangle made of 16 small equilateral triangles

Explanation:

Part (b)

Step1: Analyze the first triangle

The first triangle has 1 small equilateral triangle, so its area is 1 square unit (given).

Step2: Analyze the second triangle

The second triangle is made up of \(1 + 3 = 4\) small equilateral triangles (since it's a larger equilateral triangle divided into 4 small ones). So area is \(4\times1 = 4\) square units.

Step3: Analyze the third triangle

The third triangle has a base with 3 small triangles, so the number of small triangles is \(1 + 3 + 5 = 9\) (sum of first 3 odd numbers). Area is \(9\times1 = 9\) square units.

Step4: Analyze the fourth triangle

The fourth triangle has a base with 4 small triangles, so the number of small triangles is \(1 + 3 + 5 + 7 = 16\) (sum of first 4 odd numbers). Area is \(16\times1 = 16\) square units.

When triangles are made from equilateral triangles, all sides of the larger triangle are composed of equal - length sides of the small equilateral triangles. By the definition of an equilateral triangle (a triangle with all three sides of equal length), any triangle formed by combining equilateral triangles will have all three sides equal, so it is equilateral.

Step1: Identify the pattern in areas

The areas of the triangles are \(1 = 1^2\), \(4 = 2^2\), \(9 = 3^2\), \(16 = 4^2\). So the pattern is that the area of the \(n\)th triangle is \(n^2\) square units.

Step2: Predict the area of the seventh triangle

For the seventh triangle, \(n = 7\). Using the pattern \(A=n^2\), where \(A\) is the area and \(n\) is the triangle number. So the area is \(7^2=49\) square units.

Answer:

The areas of the first four triangles are 1 square unit, 4 square units, 9 square units, and 16 square units.

Part (a)