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Wize High School Grade 10 Math Textbook > Analytic Geometry: Geometric Properties

Properties of Quadrilaterals

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Example: Properties of Quadrilaterals

A quadrilateral is given by the vertices P(1,2), Q(5,3), R(3,−1), S(−1,1)P(1,2),~Q(5,3),~R(3,-1),~S(-1,1). We connect the midpoints of each side of this quadrilateral to form a new quadrilateral. Classify this new quadrilateral.

Midpoints:
MPQ=(x1+x22,y1+y22)=(1+52,2+32)=(3,52)MQR=(x1+x22,y1+y22)=(5+32,3+(−1)2)=(4,1)\begin{array}{c|c|} \begin{array}{rl} M_{PQ}=&\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)\\[1em] =&\left(\dfrac{1+5}{2},\dfrac{2+3}{2}\right)\\[1em] =&\left(3,\dfrac{5}{2}\right)\\[1em] \end{array} & \begin{array}{rl} M_{QR}=&\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)\\[1em] =&\left(\dfrac{5+3}{2},\dfrac{3+(-1)}{2}\right)\\[1em] =&\left(4,1\right)\\[1em] \end{array} \end{array}

MRS=(x1+x22,y1+y22)=(3+(−1)2,(−1)+12)=(1,0)MSP=(x1+x22,y1+y22)=((−1)+12,1+22)=(0,32)\begin{array}{c|c|} \begin{array}{rl} M_{RS}=&\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)\\[1em] =&\left(\dfrac{3+(-1)}{2},\dfrac{(-1)+1}{2}\right)\\[1em] =&\left(1,0\right)\\[1em] \end{array} & \begin{array}{rl} M_{SP}=&\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)\\[1em] =&\left(\dfrac{(-1)+1}{2},\dfrac{1+2}{2}\right)\\[1em] =&\left(0,\dfrac{3}{2}\right)\\[1em] \end{array} \end{array}

Side Lengths of new quadrilateral:
Side1 MPQMQR=(x2−x1)2+(y2−y1)2=(4−3)2+(1−52)2=(1)2+(−32)2=1+(94)=(134)Side2 MQRMRS=(x2−x1)2+(y2−y1)2=(1−4)2+(0−1)2=(−3)2+(−1)2=9+1=10\begin{array}{c|c|} \begin{array}{rl} \text{Side1 }M_{PQ}M_{QR}=&\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\[1em] =&\sqrt{\left(4-3\right)^2+\left(1-\dfrac{5}{2}\right)^2}\\[1em] =&\sqrt{\left(1\right)^2+\left(-\dfrac{3}{2}\right)^2}\\[1em] =&\sqrt{1+\left(\dfrac{9}{4}\right)}\\[1em] =&\colorThree{\sqrt{\left(\dfrac{13}{4}\right)}}\\[1em] \end{array} & \begin{array}{rl} \text{Side2 }M_{QR}M_{RS}=&\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\[1em] =&\sqrt{\left(1-4\right)^2+\left(0-1\right)^2}\\[1em] =&\sqrt{\left(-3\right)^2+\left(-1\right)^2}\\[1em] =&\sqrt{9+1}\\[1em] =&\colorThree{\sqrt{10}}\\[1em] \\[1em] \\ \end{array} \end{array}

Side3 MRSMSP=(x2−x1)2+(y2−y1)2=(0−1)2+(32−0)2=(−1)2+(32)2=1+(94)=(134)Side4 MSPMPQ=(x2−x1)2+(y2−y1)2=(3−0)2+(52−32)2=(3)2+(1)2=9+1=10\begin{array}{c|c|} \begin{array}{rl} \text{Side3 }M_{RS}M_{SP}=&\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\[1em] =&\sqrt{\left(0-1\right)^2+\left(\dfrac{3}{2}-0\right)^2}\\[1em] =&\sqrt{\left(-1\right)^2+\left(\dfrac{3}{2}\right)^2}\\[1em] =&\sqrt{1+\left(\dfrac{9}{4}\right)}\\[1em] =&\colorThree{\sqrt{\left(\dfrac{13}{4}\right)}}\\[1em] \end{array} & \begin{array}{rl} \text{Side4 }M_{SP}M_{PQ}=&\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\[1em] =&\sqrt{\left(3-0\right)^2+\left(\dfrac{5}{2}-\dfrac{3}{2}\right)^2}\\[1em] =&\sqrt{\left(3\right)^2+\left(1\right)^2}\\[1em] =&\sqrt{9+1}\\[1em] =&\colorThree{\sqrt{10}}\\[1em] \\[1em] \\ \end{array} \end{array}

There are 2 pairs of equal side lengths -- the opposite side lengths are the same

Slopes of the new quadrilateral
mside1=y2−y1x2−x1=1−524−3=−321=−32mside2=y2−y1x2−x1=0−11−4=−1−3=13mside3=y2−y1x2−x1=32−00−1=32−1=−32mside4=y2−y1x2−x1=52−323−0=13\begin{array}{c|c|c|c} \begin{array}{rl} m_\text{side1}=&\dfrac{y_2-y_1}{x_2-x_1}\\[1em] =&\dfrac{1-\dfrac{5}{2}}{4-3}\\[1em] =&\dfrac{-\dfrac{3}{2}}{1}\\[1em] =&\colorThree{-\dfrac{3}{2}}\\[1em] \end{array} & \begin{array}{rl} m_\text{side2}=&\dfrac{y_2-y_1}{x_2-x_1}\\[1em] =&\dfrac{0-1}{1-4}\\[1em] =&\dfrac{-1}{-3}\\[1em] =&\colorThree{\dfrac{1}{3}}\\[1em] \\[1em] \end{array} & \begin{array}{rl} m_\text{side3}=&\dfrac{y_2-y_1}{x_2-x_1}\\[1em] =&\dfrac{\dfrac{3}{2}-0}{0-1}\\[1em] =&\dfrac{\dfrac{3}{2}}{-1}\\[1em] =&\colorThree{-\dfrac{3}{2}}\\[1em] \end{array} &\begin{array}{rl} m_\text{side4}=&\dfrac{y_2-y_1}{x_2-x_1}\\[1em] =&\dfrac{\dfrac{5}{2}-\dfrac{3}{2}}{3-0}\\[1em] =&\colorThree{\dfrac{1}{3}}\\[1em] \\[3em] \end{array} \end{array}

The opposite sides are parallel, and there are no right angles.

Therefore, this new quadrilateral is a parallelogram.

Practice: Properties of Quadrilaterals

A rectangle is given by the vertices A(1,3), B(5,−1), C(3,−3), D(−1,1)A(1,3),~B(5,-1),~C(3,-3),~D(-1,1). What can you conclude about the diagonals of this rectangle?

Select all that apply.
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Practice: Properties of Quadrilaterals

A quadrilateral has vertices A(−2,2), B(0,2), C(3,0), D(−3,0)A(-2,2),~B(0,2),~C(3,0),~D(-3,0).

a) Show that this is trapezoid by finding the lengths and slopes of each side.

b) Show that the line segment joining the midpoints of the non-parallel sides of the trapezoid is parallel to the parallel sides of the trapezoid.

c) Show that the line segment joining the midpoints of the non-parallel sides of the trapezod has a length equal to the average of the lengths of the parallel sides.