Question

Use slopes to show that the quadrilateral

whose vertices are $(-1,0),(2,3),(1,-2)$and $(4,1)$is a rectangle.

Short answer

The given quadrilateral is a rectangle.

Step-by-step solution

Step 1. Given information 

A quadrilateral with vertices$(-1,0),(2,3),(1,-2)$and$(4,1)$

Step 2. Required to find 

Show that the quadrilateral is a rectangle.

For this, show that slopes of the lines making opposite sides of quadrilateral are equal. Adjacent sides are perpendicular.

Step 3. Finding the slopes of the sides of quadrilateral

Slope, $m=\frac{y_2-y_1}{x_2-x_1}$

Slope of line joining vertices $(-1,0)$and $(1,-2)$is: $m_1=\frac{-2-0}{1-(-1)}$

$m_1=-1$

Slope of line joining vertices role="math" localid="1646111060060" $(1,-2)$and role="math" localid="1646111064854" $(4,1)$is: role="math" localid="1646111069864" $m_2=\frac{1-(-2)}{4-1}$

$m_2=1$

Slope of line joining vertices role="math" localid="1646111095728" $(4,1)$and role="math" localid="1646111105643" $(2,3)$is: role="math" localid="1646111199853" $m_3=\frac{3-1}{2-4}$

role="math" localid="1646111214318" $m_3=-1$

Slope of line joining vertices $(2,3)$and $(-1,0)$is: $m_4=\frac{0-3}{-1-2}$

$m_4=1$

Step 4. Showing opposite sides are parallel

As, $m_1=m_3$and $m_2=m_4$, the opposite sides of quadrilateral have same slopes. So, they are parallel.

Step 5. Showing adjacent sides are perpendicular

As $m_1\times m_2=-1,m_2\times m_3=-1,m_3\times m_4=-1,m_4\times m_1=-1$,the adjacent sides are perpendicular.