Many properties of quadrilaterals can either be verified or proved using coordinate geometry.
There is a range of established formulas that become useful in this endeavor. You may wish to go back to a previous lesson to review each one.
Show that the quadrilateral with vertices given by $P\left(2,3\right),Q\left(3,6\right),R\left(6,8\right),S\left(5,5\right)$P(2,3),Q(3,6),R(6,8),S(5,5) is a parallelogram.
A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
Think: If opposite sides are parallel then they must have the same slope. We need $\overline{PQ}\parallel\overline{RS}$PQ∥RS and $\overline{QR}\parallel\overline{PS}$QR∥PS
Do: We can check the slopes of $\overline{PQ}$PQ and $\overline{RS}$RS and the slopes of the line segments $\overline{QR}$QR and $\overline{PS}$PS.
$m_{\overline{PQ}}=\frac{6-3}{3-2}=3$mPQ=6−33−2=3
$m_{\overline{RS}}=\frac{5-8}{5-6}=3$mRS=5−85−6=3
$\overline{PQ}\parallel\overline{RS}$PQ∥RS
$m_{\overline{QR}}=\frac{8-6}{6-3}=\frac{2}{3}$mQR=8−66−3=23
$m_{\overline{PS}}=\frac{5-3}{5-2}=\frac{2}{3}$mPS=5−35−2=23
$\overline{QR}\parallel\overline{PS}$QR∥PS
Reflect: Both pairs of opposite sides are parallel. Hence, the quadrilateral is a parallelogram.
Prove that the quadrilateral with vertices $A\left(4,9\right),B\left(5,13\right),C\left(9,14\right),D\left(8,10\right)$A(4,9),B(5,13),C(9,14),D(8,10) is a rhombus.
Think: A rhombus must have all sides of equal length and opposite sides must be parallel. It does not need to be a square, so the adjacent sides do not need to be perpendicular.
Do: Let's look at distances first, to ensure that they are all the same. We find:
$\overline{AB}=\sqrt{1^2+4^2}=\sqrt{17}$AB=√12+42=√17
$\overline{BC}=\sqrt{4^2+1^2}=\sqrt{17}$BC=√42+12=√17
$\overline{CD}=\sqrt{1^2+4^2}=\sqrt{17}$CD=√12+42=√17
$\overline{DA}=\sqrt{4^2+1^2}=\sqrt{17}$DA=√42+12=√17
Next, let's calculate slope to ensure that opposite sides are parallel,
$m_{\overline{AB}}=\frac{4}{1}=4$mAB=41=4
$m_{\overline{CD}}=\frac{-4}{-1}=4$mCD=−4−1=4
so $m_{\overline{AB}}\parallel m_{\overline{CD}}$mAB∥mCD
$m_{\overline{BC}}=\frac{1}{4}$mBC=14
$m_{\overline{DA}}=\frac{1}{4}$mDA=14
so $m_{\overline{BC}}\parallel m_{\overline{DA}}$mBC∥mDA
Reflect: The opposite side pairs are parallel and all sides are the same length, so this is indeed a rhombus.
The points given represent three vertices of a parallelogram. What is the fourth vertex if it is known to be in the 2nd quadrant?
Coordinates $=$=$\left(\editable{},\editable{}\right)$(,)
Given Line P: $y=-6x-4$y=−6x−4, Line Q: $y=\frac{x}{6}+6$y=x6+6, Line R: $y=-6x-1$y=−6x−1 and Line S: $y=\frac{x}{6}+1$y=x6+1.
Complete the following:
$m$mP = $\editable{}$
$m$mQ = $\editable{}$
$m$mP x $m$mQ = $\editable{}$
Complete the following:
$m$mQ = $\editable{}$
$m$mR = $\editable{}$
$m$mQ x $m$mR = $\editable{}$
Complete the following:
$m$mR = $\editable{}$
$m$mS = $\editable{}$
$m$mR x $m$mS = $\editable{}$
Complete the following:
$m$mS = $\editable{}$
$m$mP = $\editable{}$
$m$mS x $m$mP = $\editable{}$
What type of quadrilateral is formed by lines: P, Q, R, and S?
Trapezoid
Rectangle
Rhombus
Parallelogram
The four points $A$A$\left(3,4\right)$(3,4), $B$B$\left(6,2\right)$(6,2), $C$C$\left(5,-3\right)$(5,−3) , and $D$D$\left(-4,3\right)$(−4,3) are the vertices of a quadrilateral.
Plot the quadrilateral.
Find the slope of $\overline{AB}$AB.
Find the slope of $\overline{BC}$BC.
Find the slope of $\overline{CD}$CD.
Find the slope of $\overline{DA}$DA.
Which segments are parallel?
$\overline{AB}$AB and $\overline{BC}$BC
$\overline{BC}$BC and $\overline{CD}$CD
$\overline{AB}$AB and $\overline{CD}$CD
$\overline{DA}$DA and $\overline{CD}$CD
What type of quadrilateral is $ABCD$ABCD ? Choose the most precise answer.
Rectangle
Parallelogram
Trapezoid
Rhombus