Question

figure lmno is located at l (1, 1), m (2, 3), n (4, 3), and o (5, 1). use coordinate geometry to best classify quadrilateral lmno. trapezoid square parallelogram kite

Explanation:

Step1: Calculate the slopes of the sides

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
For side \(LM\): \(L(1,1)\) and \(M(2,3)\), \(m_{LM}=\frac{3 - 1}{2 - 1}=2\).
For side \(MN\): \(M(2,3)\) and \(N(4,3)\), \(m_{MN}=\frac{3 - 3}{4 - 2}=0\).
For side \(NO\): \(N(4,3)\) and \(O(5,1)\), \(m_{NO}=\frac{1 - 3}{5 - 4}=-2\).
For side \(OL\): \(O(5,1)\) and \(L(1,1)\), \(m_{OL}=\frac{1 - 1}{1 - 5}=0\).

Step2: Check the properties of parallelogram

In a parallelogram, opposite sides are parallel.
\(m_{LM}=2\) and \(m_{NO}=-2\) (not parallel). \(m_{MN}=0\) and \(m_{OL}=0\) (parallel).
Let's use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
\(d_{LM}=\sqrt{(2 - 1)^2+(3 - 1)^2}=\sqrt{1 + 4}=\sqrt{5}\).
\(d_{MN}=\sqrt{(4 - 2)^2+(3 - 3)^2}=2\).
\(d_{NO}=\sqrt{(5 - 4)^2+(1 - 3)^2}=\sqrt{1 + 4}=\sqrt{5}\).
\(d_{OL}=\sqrt{(1 - 5)^2+(1 - 1)^2}=4\).

Step3: Check the properties of trapezoid

A trapezoid has at least one pair of parallel sides. Here \(MN\parallel OL\) (since \(m_{MN}=m_{OL} = 0\)).

Answer:

Trapezoid