Question

name:
directions: now graph and label quadrilateral to prove that it is parallelogram

1. graph and label quadrilateral smrt with vertices s(1, 2), m(2, -1), r(8, 1) and t(7, 4)

b) prove that quadrilateral smrt is a rectangle.

Explanation:

Step1: Plot vertices on coordinate plane

Mark points $S(1, 2)$, $M(2, -1)$, $R(8, 1)$, $T(7, 4)$ on the grid, then connect them in order to form quadrilateral $SMRT$.

Step2: Calculate side slopes (slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$)

Slope of $SM$: $m_{SM}=\frac{-1-2}{2-1}=-3$
Slope of $MR$: $m_{MR}=\frac{1-(-1)}{8-2}=\frac{2}{6}=\frac{1}{3}$
Slope of $RT$: $m_{RT}=\frac{4-1}{7-8}=\frac{3}{-1}=-3$
Slope of $TS$: $m_{TS}=\frac{2-4}{1-7}=\frac{-2}{-6}=\frac{1}{3}$

Step3: Verify parallel sides

$m_{SM}=m_{RT}=-3$, so $SM \parallel RT$; $m_{MR}=m_{TS}=\frac{1}{3}$, so $MR \parallel TS$. Thus, $SMRT$ is a parallelogram.

Step4: Check perpendicular sides

For perpendicular lines, $m_1 \times m_2 = -1$.
$m_{SM} \times m_{MR} = -3 \times \frac{1}{3} = -1$, so $SM \perp MR$.

Step5: Conclude rectangle status

A parallelogram with one right angle is a rectangle.

Answer:

1. Graph: Quadrilateral $SMRT$ is formed by plotting and connecting $S(1, 2)$, $M(2, -1)$, $R(8, 1)$, $T(7, 4)$ in order on the coordinate grid.
2. Proof: Since opposite sides are parallel (equal slopes) and adjacent sides are perpendicular (product of slopes = -1), quadrilateral $SMRT$ is a rectangle.