# 11.06 Coordinate methods with quadrilaterals

Lesson

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.

• The distance formula given by $d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$d=(x2x1)2+(y2y1)2
• The slope formula $m=\frac{y_2-y_1}{x_2-x_1}$m=y2y1x2x1
• The perpendicular property $m_1*m_2=-1$m1*m2=1
• The parallel property $m_1=m_2$m1=m2
• The midpoint formula given by $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$(x1+x22,y1+y22)

#### Worked examples

##### Question 1

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 $\overline{PQ}\parallel\overline{RS}$PQRS and $\overline{QR}\parallel\overline{PS}$QRPS

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=6332=3

$m_{\overline{RS}}=\frac{5-8}{5-6}=3$mRS=5856=3

$\overline{PQ}\parallel\overline{RS}$PQRS

$m_{\overline{QR}}=\frac{8-6}{6-3}=\frac{2}{3}$mQR=8663=23

$m_{\overline{PS}}=\frac{5-3}{5-2}=\frac{2}{3}$mPS=5352=23

$\overline{QR}\parallel\overline{PS}$QRPS

Reflect: Both pairs of opposite sides are parallel. Hence, the quadrilateral is a parallelogram.

##### Question 2

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=41=4

so $m_{\overline{AB}}\parallel m_{\overline{CD}}$mABmCD

$m_{\overline{BC}}=\frac{1}{4}$mBC=14

$m_{\overline{DA}}=\frac{1}{4}$mDA=14

so $m_{\overline{BC}}\parallel m_{\overline{DA}}$mBCmDA

Reflect: The opposite side pairs are parallel and all sides are the same length, so this is indeed a rhombus.

#### Practice questions

##### Question 3

The points given represent three vertices of a parallelogram. What is the fourth vertex if it is known to be in the 2nd quadrant?

1. Coordinates $=$=$\left(\editable{},\editable{}\right)$(,)

##### Question 4

Given Line P: $y=-6x-4$y=6x4, Line Q: $y=\frac{x}{6}+6$y=x6+6, Line R: $y=-6x-1$y=6x1 and Line S: $y=\frac{x}{6}+1$y=x6+1.

1. Complete the following:

$m$mP = $\editable{}$

$m$mQ = $\editable{}$

$m$mP x $m$mQ = $\editable{}$

2. Complete the following:

$m$mQ = $\editable{}$

$m$mR = $\editable{}$

$m$mQ x $m$mR = $\editable{}$

3. Complete the following:

$m$mR = $\editable{}$

$m$mS = $\editable{}$

$m$mR x $m$mS = $\editable{}$

4. Complete the following:

$m$mS = $\editable{}$

$m$mP = $\editable{}$

$m$mS x $m$mP = $\editable{}$

5. What type of quadrilateral is formed by lines: P, Q, R, and S?

Trapezoid

A

Rectangle

B

Rhombus

C

Parallelogram

D

Trapezoid

A

Rectangle

B

Rhombus

C

Parallelogram

D

##### Question 5

The four points $A$A$\left(3,4\right)$(3,4), $B$B$\left(6,2\right)$(6,2), $C$C$\left(-4,3\right)$(4,3) , and $D$D$\left(5,-3\right)$(5,3) are the vertices of a quadrilateral.

2. Find the slope of $\overline{AB}$AB.

3. Find the slope of  $\overline{BC}$BC.

4. Find the slope of $\overline{CD}$CD.

5. Find the slope of $\overline{DA}$DA.

6. Which segments are parallel?

$\overline{AB}$AB and $\overline{BC}$BC

A

$\overline{BC}$BC and $\overline{CD}$CD

B

$\overline{AB}$AB and $\overline{CD}$CD

C

$\overline{DA}$DA and $\overline{CD}$CD

D

$\overline{AB}$AB and $\overline{BC}$BC

A

$\overline{BC}$BC and $\overline{CD}$CD

B

$\overline{AB}$AB and $\overline{CD}$CD

C

$\overline{DA}$DA and $\overline{CD}$CD

D
7. What type of quadrilateral is $ABCD$ABCD ?  Choose the most precise answer.

Rectangle

A

Parallelogram

B

Trapezoid

C

Rhombus

D

Rectangle

A

Parallelogram

B

Trapezoid

C

Rhombus

D

### Outcomes

#### II.G.GPE.4

Use coordinates to prove simple geometric theorems algebraically.