Each successive figure below is made of small triangles like the first one in the sequence. Conjecture the number of small triangles needed to make a. the 100th figure? b. the nth figure? The first figure is an equilateral triangle. In the second figure, lines parallel to each side bisect the other two sides, forming 4 equilateral triangles. In the third figure, two lines parallel to each side of the triangle trisect each side and form a total of 9 small equilateral triangles.
a. How many small triangles are needed to make the 100th figure?
It would require ? small triangles
b. How many small triangles are needed to make the nth figure?
It would require _ small triangles.
The Correct Answer and Explanation is:
Let’s analyze the problem step-by-step.
Given:
- The 1st figure is a single equilateral triangle → 1 small triangle
- The 2nd figure has each side bisected (divided into 2 equal parts), creating 4 small equilateral triangles.
- The 3rd figure has each side trisected (divided into 3 equal parts), creating 9 small equilateral triangles.
What is the pattern?
From the description:
- 1st figure: 1 small triangle (side divided into 1 part)
- 2nd figure: 4 small triangles (side divided into 2 parts)
- 3rd figure: 9 small triangles (side divided into 3 parts)
So for the nth figure, the sides are divided into n equal parts, forming a grid of small equilateral triangles.
The number of small triangles in the nth figure
Notice that:
- 1st figure → $1^2 = 1$
- 2nd figure → $2^2 = 4$
- 3rd figure → $3^2 = 9$
So the number of small triangles is the square of the number of divisions per side, which is n².
Answers:
a. The 100th figure would require:
$$
100^2 = 10,000
$$
small triangles.
b. The nth figure would require:
$$
n^2
$$
small triangles.
Explanation (300+ words):
This problem involves geometric subdivision of an equilateral triangle by dividing each side into equal segments and drawing lines parallel to the triangle’s sides to form smaller equilateral triangles.
- The first figure is just one large equilateral triangle. So the count of small triangles is clearly 1.
- The second figure divides each side into 2 equal parts, so two parallel lines are drawn inside the triangle, intersecting to form smaller triangles. When you divide a side into 2 parts, you essentially create a grid that fits $2 \times 2 = 4$ small equilateral triangles inside the larger triangle.
- The third figure divides each side into 3 equal parts, and lines parallel to the sides create a grid of $3 \times 3 = 9$ small triangles.
From these observations, the pattern emerges clearly: when you divide each side into n parts, the number of small triangles inside the big triangle is $n^2$.
Why does this happen?
The division into equal parts creates smaller segments, and the lines drawn parallel to each side form a grid of smaller equilateral triangles, not just rectangles or squares. Because the original shape is an equilateral triangle, the smaller triangles formed have equal sides and are arranged in a triangular grid pattern. The number of small triangles corresponds to the total number of these smaller equilateral triangles fitting inside the large triangle.
Each subdivision step increases the number of small triangles quadratically because the subdivision affects both dimensions — essentially, the triangle is divided “vertically” and “horizontally” by lines parallel to its sides, leading to $n^2$ smaller triangles.
Therefore:
- For $n = 1$, small triangles = $1^2 = 1$
- For $n = 2$, small triangles = $2^2 = 4$
- For $n = 3$, small triangles = $3^2 = 9$
And so on.
Hence, the formula for the number of small triangles in the nth figure is:
$$
\boxed{n^2}
$$
So for the 100th figure:
$$
100^2 = 10,000
$$
small triangles are required.
If you want, I can help illustrate the pattern or create a visual to make this clearer!