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

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Consider quadrilateral $$LMNO$$ with vertices  
$$
L(1,1),\quad M(4,4),\quad N(7,1),\quad O(4,-2).
$$

**Step 1. Compute the side vectors and their lengths.**

- **Side $$LM$$:**  
  $$
  \overrightarrow{LM} = M - L = (4-1,\, 4-1) = (3,\, 3).
  $$
  Length:  
  $$
  LM = \sqrt{3^2 + 3^2} = \sqrt{9+9} = \sqrt{18} = 3\sqrt{2}.
  $$

- **Side $$MN$$:**  
  $$
  \overrightarrow{MN} = N - M = (7-4,\, 1-4) = (3,\,-3).
  $$
  Length:  
  $$
  MN = \sqrt{3^2 + (-3)^2} = \sqrt{9+9} = 3\sqrt{2}.
  $$

- **Side $$NO$$:**  
  $$
  \overrightarrow{NO} = O - N = (4-7,\,-2-1) = (-3,\,-3).
  $$
  Length:  
  $$
  NO = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9+9} = 3\sqrt{2}.
  $$

- **Side $$OL$$:**  
  $$
  \overrightarrow{OL} = L - O = (1-4,\, 1-(-2)) = (-3,\, 3).
  $$
  Length:  
  $$
  OL = \sqrt{(-3)^2 + 3^2} = \sqrt{9+9} = 3\sqrt{2}.
  $$

All sides are congruent.

**Step 2. Check for right angles.**

Calculate the dot product between adjacent side vectors:

- Between $$ \overrightarrow{LM} = (3,3) $$ and $$ \overrightarrow{MN} = (3,-3) $$:
  $$
  (3)(3) + (3)(-3) = 9 - 9 = 0.
  $$
A zero dot product indicates a right angle.

Since one pair of adjacent sides is perpendicular and the quadrilateral has all sides congruent, we can conclude that all angles are $$90^\circ$$.

**Conclusion:**

The quadrilateral has four congruent sides and four right angles, which is the definition of a square.

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

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