Question
Raji Draw the square with the side length of X + 5 centimeters. Ava draws a regular hexagon with a side length of X - 1 centimeter. The two figures have the same perimeter. Write an equation that allows you to solve for X. What is the value of X?
Answers
GPT-4o mini
To find the value of \( X \), let's first express the perimeters of both the square and the regular hexagon in terms of \( X \).
1. **Perimeter of the square**:
The formula for the perimeter \( P \) of a square with side length \( s \) is:
\[
P = 4s
\]
For the square with side length \( X + 5 \):
\[
P_{\text{square}} = 4(X + 5) = 4X + 20
\]
2. **Perimeter of the hexagon**:
The formula for the perimeter \( P \) of a regular hexagon with side length \( s \) is:
\[
P = 6s
\]
For the hexagon with side length \( X - 1 \):
\[
P_{\text{hexagon}} = 6(X - 1) = 6X - 6
\]
Since the two figures have the same perimeter, we can set the equations equal to each other:
\[
4X + 20 = 6X - 6
\]
Now, let's solve for \( X \):
1. Rearrange the equation:
\[
20 + 6 = 6X - 4X
\]
\[
26 = 2X
\]
2. Divide both sides by 2:
\[
X = 13
\]
Thus, the value of \( X \) is \( \boxed{13} \).
1. **Perimeter of the square**:
The formula for the perimeter \( P \) of a square with side length \( s \) is:
\[
P = 4s
\]
For the square with side length \( X + 5 \):
\[
P_{\text{square}} = 4(X + 5) = 4X + 20
\]
2. **Perimeter of the hexagon**:
The formula for the perimeter \( P \) of a regular hexagon with side length \( s \) is:
\[
P = 6s
\]
For the hexagon with side length \( X - 1 \):
\[
P_{\text{hexagon}} = 6(X - 1) = 6X - 6
\]
Since the two figures have the same perimeter, we can set the equations equal to each other:
\[
4X + 20 = 6X - 6
\]
Now, let's solve for \( X \):
1. Rearrange the equation:
\[
20 + 6 = 6X - 4X
\]
\[
26 = 2X
\]
2. Divide both sides by 2:
\[
X = 13
\]
Thus, the value of \( X \) is \( \boxed{13} \).