Question

What is the mathematical relationship between the area of the original square and the area of the square formed by joining the midpoints of its sides?

Answer 1

To find the relationship between the area of an original square and the area of the square formed by joining the midpoints of its sides, let's first denote the side length of the original square as \( s \).

The area \( A \) of the original square can be calculated as:

\[
A = s^2
\]

Now, let's consider the new square formed by joining the midpoints of the original square's sides. The midpoints of the sides of the square will be located at the following coordinates if we place the original square in a coordinate system:

- Bottom-left corner: \( (0, 0) \)
- Bottom-right corner: \( (s, 0) \)
- Top-right corner: \( (s, s) \)
- Top-left corner: \( (0, s) \)

The midpoints are located at:

- Midpoint of the bottom side: \( \left( \frac{s}{2}, 0 \right) \)
- Midpoint of the right side: \( \left( s, \frac{s}{2} \right) \)
- Midpoint of the top side: \( \left( \frac{s}{2}, s \right) \)
- Midpoint of the left side: \( \left( 0, \frac{s}{2} \right) \)

For clarity, let's denote the midpoints as points \( A, B, C, D \):

- \( A = \left( \frac{s}{2}, 0 \right) \)
- \( B = \left( s, \frac{s}{2} \right) \)
- \( C = \left( \frac{s}{2}, s \right) \)
- \( D = \left( 0, \frac{s}{2} \right) \)

Next, we need to determine the distance between two adjacent midpoints to find the side length of the new square. Let's calculate the distance between points \( A \) and \( B \):

The distance \( d \) can be calculated using the distance formula:

\[
d = \sqrt{ \left( \frac{s}{2} - s \right)^2 + \left( 0 - \frac{s}{2} \right)^2 }
\]

Calculating each part:

\[
\frac{s}{2} - s = -\frac{s}{2} \quad \text{and} \quad 0 - \frac{s}{2} = -\frac{s}{2}
\]

So,

\[
d = \sqrt{ \left( -\frac{s}{2} \right)^2 + \left( -\frac{s}{2} \right)^2 } = \sqrt{ \frac{s^2}{4} + \frac{s^2}{4} } = \sqrt{ \frac{s^2}{2} } = \frac{s}{\sqrt{2}}
\]

The side length of the new square is \( \frac{s}{\sqrt{2}} \).

Now, we can calculate the area \( A' \) of the new square:

\[
A' = \left( \frac{s}{\sqrt{2}} \right)^2 = \frac{s^2}{2}
\]

Finally, we can express the relationship between the area of the original square and the area of the new square:

\[
\text{Area of new square} = \frac{1}{2} \text{ (Area of original square)}
\]

Thus, the mathematical relationship between the area of the original square and the area of the square formed by joining the midpoints of its sides is:

\[
\text{Area of new square} = \frac{1}{2} \text{Area of original square}
\]