What shape is formed when you join the midpoints of the sides of rectangle RECT? Let EC = 16 and CT = 10. Is this true for any rectangle?

Steps to Solve

1. Identify the midpoints of the rectangle sides

Given rectangle RECT, label the vertices as follows:

• $R(0, 0)$
• $E(16, 0)$
• $C(16, 10)$
• $T(0, 10)$

Now, find the midpoints of each side:

• Midpoint of side RE: $M_{RE} = \left(\frac{0 + 16}{2}, \frac{0 + 0}{2}\right) = (8, 0)$
• Midpoint of side EC: $M_{EC} = \left(\frac{16 + 16}{2}, \frac{0 + 10}{2}\right) = (16, 5)$
• Midpoint of side CT: $M_{CT} = \left(\frac{16 + 0}{2}, \frac{10 + 10}{2}\right) = (8, 10)$
• Midpoint of side TR: $M_{TR} = \left(\frac{0 + 0}{2}, \frac{10 + 0}{2}\right) = (0, 5)$

2. Connect the midpoints

Next, connect the midpoints $M_{RE}$, $M_{EC}$, $M_{CT}$, and $M_{TR}$. This will create a new shape.

3. Analyze the shape formed

To analyze the shape, consider the coordinates of the midpoints:

• $M_{RE} = (8, 0)$
• $M_{EC} = (16, 5)$
• $M_{CT} = (8, 10)$
• $M_{TR} = (0, 5)$

The shape formed by connecting these midpoints is a parallelogram since opposite sides (like $M_{RE}M_{CT}$ and $M_{EC}M_{TR}$) are equal in length and parallel.

4. Determine if this is true for any rectangle

Yes, this property is true for any rectangle. The midpoints will always create a parallelogram, specifically a rectangle or a square, depending on the original rectangle's proportions.