Question

classifying a quadrilateral on the coordinate plane
determine whether quadrilateral abcd with vertices
a(-4,-5), b(-3,0), c(0,2), and d(5,1) is a trapezoid.
step 1: find the slope of ab. the slope of ab is
step 2: find the slope of dc. the slope of dc is
step 3: find the slope of bc. the slope of bc is
step 4: find the slope of ad. the slope ad is
the quadrilateral is a trapezoid because

Explanation:

Step1: Calculate the slope of \(AB\)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For points \(A(-4,-5)\) and \(B(-3,0)\), \(m_{AB}=\frac{0-(-5)}{-3 - (-4)}=\frac{0 + 5}{-3+4}=\frac{5}{1}=5\).

Step2: Calculate the slope of \(DC\)

For points \(D(5,1)\) and \(C(0,2)\), \(m_{DC}=\frac{2 - 1}{0 - 5}=\frac{1}{-5}=-\frac{1}{5}\).

Step3: Calculate the slope of \(BC\)

For points \(B(-3,0)\) and \(C(0,2)\), \(m_{BC}=\frac{2-0}{0-(-3)}=\frac{2}{3}\).

Step4: Calculate the slope of \(AD\)

For points \(A(-4,-5)\) and \(D(5,1)\), \(m_{AD}=\frac{1-(-5)}{5-(-4)}=\frac{1 + 5}{5 + 4}=\frac{6}{9}=\frac{2}{3}\).

Since \(m_{BC}=m_{AD}=\frac{2}{3}\), and a trapezoid is a quadrilateral with at least one pair of parallel sides (parallel lines have equal slopes).

Answer:

Slope of \(AB\): \(5\); Slope of \(DC\): \(-\frac{1}{5}\); Slope of \(BC\): \(\frac{2}{3}\); Slope of \(AD\): \(\frac{2}{3}\); The quadrilateral is a trapezoid because \(BC\parallel AD\) (or one pair of sides \(BC\) and \(AD\) are parallel).