Question 2 (Multiple Choice Worth 1 points) (04.01 MC) Figure LMNO is located at $$ \mathrm{L}(1,1), \mathrm{M}(4,4), \mathrm{N}(7,1) $$, and $$ \mathrm{O}(4,-2) $$. Use coordinate geometry to best classify quadrilateral LMNO. O Kite O Parallelogram O Square Trapezoid Question 3 ?Multiono

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1. List the vertices:  
   $$
   \mathrm{L}(1,1), \quad \mathrm{M}(4,4), \quad \mathrm{N}(7,1), \quad \mathrm{O}(4,-2)
   $$

2. Compute the side lengths using the distance formula.  
   $$
   d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   $$

- Side $$\overline{LM}$$:
     $$
     \sqrt{(4-1)^2 + (4-1)^2} = \sqrt{3^2 + 3^2} = \sqrt{9+9} = \sqrt{18} = 3\sqrt{2}
     $$

- Side $$\overline{MN}$$:
     $$
     \sqrt{(7-4)^2 + (1-4)^2} = \sqrt{3^2 + (-3)^2} = \sqrt{9+9} = 3\sqrt{2}
     $$

- Side $$\overline{NO}$$:
     $$
     \sqrt{(4-7)^2 + (-2-1)^2} = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9+9} = 3\sqrt{2}
     $$

- Side $$\overline{OL}$$:
     $$
     \sqrt{(1-4)^2 + (1-(-2))^2} = \sqrt{(-3)^2 + (3)^2} = \sqrt{9+9} = 3\sqrt{2}
     $$

All four sides are equal in length.

3. Compute the slopes of consecutive sides to check for perpendicularity.  
   The slope formula is  
   $$
   m = \frac{y_2 - y_1}{x_2 - x_1}
   $$

- Slope of $$\overline{LM}$$:
     $$
     m_{LM} = \frac{4-1}{4-1} = \frac{3}{3} = 1
     $$

- Slope of $$\overline{MN}$$:
     $$
     m_{MN} = \frac{1-4}{7-4} = \frac{-3}{3} = -1
     $$

- Slope of $$\overline{NO}$$:
     $$
     m_{NO} = \frac{-2-1}{4-7} = \frac{-3}{-3} = 1
     $$

- Slope of $$\overline{OL}$$:
     $$
     m_{OL} = \frac{1-(-2)}{1-4} = \frac{3}{-3} = -1
     $$

Since the product of the slopes for adjacent sides is  
   $$
   1 \cdot (-1) = -1,
   $$  
   each adjacent pair is perpendicular.

4. With four equal sides and all angles measuring $$90^\circ$$, the quadrilateral is a square.

$$
\boxed{\text{Square}}
$$
Quadrilateral LMNO is a square.

Question 2 (Multiple Choice Worth 1 points) (04.01 MC) Figure LMNO is located at \( \mathrm{L}(1,1), \mathrm{M}(4,4), \mathrm{N}(7,1) \), and \( \mathrm{O}(4,-2) \). Use coordinate geometry to best classify quadrilateral LMNO. O Kite O Parallelogram O Square Trapezoid Question 3 ?Multiono

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