Find the coordinates of the orthocenter of the triangle with the vertices $$ L(8,0), M(10,8) $$, and $$ N(14,0) $$.

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To find the orthocenter of the triangle with vertices $$ L(8,0) $$, $$ M(10,8) $$, and $$ N(14,0) $$, follow these steps:

1. **Identify the Altitudes:**
   - The orthocenter is the intersection of the altitudes of the triangle.
   - An altitude is a perpendicular line from a vertex to the opposite side.

2. **Find the Equation of the First Altitude (from $$ M $$):**
   - **Side $$ LN $$:** Connects points $$ L(8,0) $$ and $$ N(14,0) $$.
   - Since $$ LN $$ is horizontal (both points have $$ y = 0 $$), the altitude from $$ M(10,8) $$ to $$ LN $$ is a vertical line.
   - **Equation:** $$ x = 10 $$.

3. **Find the Equation of the Second Altitude (from $$ L $$):**
   - **Side $$ MN $$:** Connects points $$ M(10,8) $$ and $$ N(14,0) $$.
   - **Slope of $$ MN $$:** 
     $$
     m_{MN} = \frac{0 - 8}{14 - 10} = \frac{-8}{4} = -2
     $$
   - The altitude from $$ L(8,0) $$ is perpendicular to $$ MN $$, so its slope is the negative reciprocal:
     $$
     m_{\text{altitude}} = \frac{1}{2}
     $$
   - **Equation of the altitude from $$ L $$:**
     $$
     y - 0 = \frac{1}{2}(x - 8) \implies y = \frac{1}{2}x - 4
     $$

4. **Determine the Orthocenter:**
   - Find the intersection of the two altitudes $$ x = 10 $$ and $$ y = \frac{1}{2}x - 4 $$.
   - Substitute $$ x = 10 $$ into the second equation:
     $$
     y = \frac{1}{2}(10) - 4 = 5 - 4 = 1
     $$
   - **Orthocenter Coordinates:** $$ (10, 1) $$.

**Answer:** The orthocenter of the triangle is at the point $$ (10,\ 1) $$.
The orthocenter of the triangle is at $$ (10, 1) $$.

Find the coordinates of the orthocenter of the triangle with the vertices \( L(8,0), M(10,8) \), and \( N(14,0) \).

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