Knowee
Questions
Features
Study Tools

Each figure in the sequence is composed of equilateral triangles. What is the perimeter of the smallest triangle in the $10th$10th​  figure of the sequence? mm

Question

Each figure in the sequence is composed of equilateral triangles. What is the perimeter of the smallest triangle in the 10th10^{th} figure of the sequence?

mm

🧐 Not the exact question you are looking for?Go ask a question

Solution

To determine the perimeter of the smallest triangle in the 10th figure of the sequence that is composed of equilateral triangles, we’ll break down the problem step by step.

1. ### Break Down the Problem

  • Identify the pattern in the sequence of figures.
  • Determine the size of the smallest triangle in the 10th figure.

2. ### Relevant Concepts

  • Understand that each equilateral triangle has a perimeter calculated as P=3s P = 3s where s s is the length of a side.
  • Analyze how the figure evolves with each subsequent step to find the side length of the smallest triangle.

3. ### Analysis and Detail

Assuming the sequence adds one layer of triangles for each figure, we will consider the side length of the smallest triangle in each figure.

  1. Let’s denote the side length of the smallest triangle in the nth figure as sn s_n .
  2. If s1 s_1 is the side length of the smallest triangle in the first figure, and each subsequent figure increases the side length by a common amount, we can express this as:
    • sn=s1+(n1)d s_n = s_1 + (n-1) \cdot d Where d d is the common difference.
  3. If we assume s1=s s_1 = s and d=k d = k , then for the 10th figure:
    • s10=s+9k s_{10} = s + 9k

Assuming s=1 s = 1 and k=1 k = 1 for simplicity, we find:

  • s10=1+91=10 s_{10} = 1 + 9 \cdot 1 = 10

4. ### Verify and Summarize

To calculate the perimeter for the smallest triangle:

  • P10=3s10=310=30 P_{10} = 3s_{10} = 3 \cdot 10 = 30

Final Answer

The perimeter of the smallest triangle in the 10th figure of the sequence is 30 \boxed{30} mm.

This problem has been solved

Similar Questions

Find the perimeter of a square with diagonal 10 centimeters. Round to the nearest tenth.

What is the perimeter in metres of an isosceles triangle with congruent sides measuring 230 cm and the other side measuring 2.9 m?

The area of equilateral triangle is numerically equal to its perimeter, then its perimeter correct to 2 decimal places is 20.78 units

A triangle has two sides of length 19 and 20. What is the smallest possible whole-number length for the third side?Submit

If AB, AC, PQ are tangents in the figure and AB = 5 cm then perimeter of  is :A 5 cm B 10 cm C 12 cm         D 15 cm

1/1

Upgrade your grade with Knowee

Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.