Trapezoid QRST has vertices Q(-1, 4), R(2, 4), S(2, 2), and T(-3, 2). Find the length of the midsegment.

Steps to Solve

1. Identify the vertices of the trapezoid

The vertices of trapezoid QRST are given as:

• ( Q(-1, 4) )
• ( R(2, 4) )
• ( S(2, 2) )
• ( T(-3, 2) )

2. Determine the midpoints of the bases

In trapezoid QRST, the bases are line segments QR and ST.

• Base QR is between points Q and R, with coordinates:

  • Midpoint ( M_1 ) is calculated as:

  $$ M_1 = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{-1 + 2}{2}, \frac{4 + 4}{2} \right) = \left( \frac{1}{2}, 4 \right) $$

• Base ST is between points S and T, with coordinates:

  • Midpoint ( M_2 ) is calculated as:

  $$ M_2 = \left( \frac{x_3 + x_4}{2}, \frac{y_3 + y_4}{2} \right) = \left( \frac{2 + (-3)}{2}, \frac{2 + 2}{2} \right) = \left( \frac{-1}{2}, 2 \right) $$

3. Calculate the length of the midsegment

The length of the midsegment ( L ) can be found by calculating the distance between the two midpoints ( M_1 ) and ( M_2 ):

• The distance formula is:

$$ L = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

Plugging in ( M_1 ) and ( M_2 ):

• ( M_1 = \left( \frac{1}{2}, 4 \right) )
• ( M_2 = \left( \frac{-1}{2}, 2 \right) )

$$ L = \sqrt{\left(\frac{-1}{2} - \frac{1}{2}\right)^2 + (2 - 4)^2} = \sqrt{\left(-1\right)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} $$