Question

1. polygon qrst has vertices q($4\frac{1}{2}$, 2), r($8\frac{1}{2}$, 2), s($8\frac{1}{2}$, $-3\frac{1}{2}$), and t($4\frac{1}{2}$, $-3\frac{1}{2}$). is polygon qrst a rectangle? justify your answer.

Explanation:

Step1: Identify coordinates of Q, R, S, T

Q($4\frac{1}{2}$, 2) = (4.5, 2), R($8\frac{1}{2}$, 2) = (8.5, 2), S($8\frac{1}{2}$, $-3\frac{1}{2}$) = (8.5, -3.5), T($4\frac{1}{2}$, $-3\frac{1}{2}$) = (4.5, -3.5)

Step2: Check opposite sides are equal

QR length: $8.5 - 4.5 = 4$; ST length: $8.5 - 4.5 = 4$
RS length: $2 - (-3.5) = 5.5$; QT length: $2 - (-3.5) = 5.5$
Opposite sides are equal.

Step3: Check right angles

Slope QR: $\frac{2-2}{8.5-4.5}=0$ (horizontal); Slope RS: $\frac{-3.5-2}{8.5-8.5}$ (undefined, vertical). Horizontal + vertical = 90°.
All consecutive sides are horizontal/vertical pairs, so all angles are right angles.

Answer:

Yes, polygon QRST is a rectangle because its opposite sides are equal and all angles are right angles.