Directions: Name each of the following in the triangle 13. The altitudes of $$ riangle \mathrm{ABC} $$ $$ \overline{14 . ext { The }} \overline{ ext { medians }} \overline{ ext { of } riangle \mathrm{AB} C} $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ 15. The centroid of $$ riangle A B C $$ point $$ \qquad $$ E is the midpoint of $$ \overline{\mathrm{AB}}, \mathrm{F} $$ is the midpoint of $$ \overline{\mathrm{BC}} $$, and G is th 16. The orthocenter of $$ riangle \mathrm{ABC} $$ point $$ \qquad $$

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Directions: Name each of the following in the triangle 13. The altitudes of $$ 	riangle \mathrm{ABC} $$ $$ \overline{14 . 	ext { The }} \overline{	ext { medians }} \overline{	ext { of } 	riangle \mathrm{AB} C} $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ 15. The centroid of $$ 	riangle A B C $$ point $$ \qquad $$ E is the midpoint of $$ \overline{\mathrm{AB}}, \mathrm{F} $$ is the midpoint of $$ \overline{\mathrm{BC}} $$, and G is th 16. The orthocenter of $$ 	riangle \mathrm{ABC} $$ point $$ \qquad $$

UpStudy AI · Accepted Answer

To name the elements in triangle $$ 	riangle ABC $$, we will identify the altitudes, medians, centroid, and orthocenter.

1. **Altitudes of $$ 	riangle ABC $$**:
   - The altitudes are the perpendicular segments from each vertex to the opposite side. 
   - For triangle $$ ABC $$:
     - The altitude from vertex $$ A $$ to side $$ BC $$ is denoted as $$ \overline{AH} $$.
     - The altitude from vertex $$ B $$ to side $$ AC $$ is denoted as $$ \overline{BI} $$.
     - The altitude from vertex $$ C $$ to side $$ AB $$ is denoted as $$ \overline{CJ} $$.

2. **Medians of $$ 	riangle ABC $$**:
   - The medians are the segments from each vertex to the midpoint of the opposite side.
   - For triangle $$ ABC $$:
     - The median from vertex $$ A $$ to side $$ BC $$ (where $$ E $$ is the midpoint of $$ \overline{BC} $$) is denoted as $$ \overline{AE} $$.
     - The median from vertex $$ B $$ to side $$ AC $$ (where $$ F $$ is the midpoint of $$ \overline{AC} $$) is denoted as $$ \overline{BF} $$.
     - The median from vertex $$ C $$ to side $$ AB $$ (where $$ G $$ is the midpoint of $$ \overline{AB} $$) is denoted as $$ \overline{CG} $$.

3. **Centroid of $$ 	riangle ABC $$**:
   - The centroid $$ G $$ is the point where the three medians intersect. It can be found by averaging the coordinates of the vertices $$ A $$, $$ B $$, and $$ C $$.
   - The centroid is denoted as point $$ G $$.

4. **Orthocenter of $$ 	riangle ABC $$**:
   - The orthocenter is the point where the three altitudes intersect.
   - It is denoted as point $$ H $$.

In summary:
- The altitudes are $$ \overline{AH} $$, $$ \overline{BI} $$, $$ \overline{CJ} $$.
- The medians are $$ \overline{AE} $$, $$ \overline{BF} $$, $$ \overline{CG} $$.
- The centroid is point $$ G $$.
- The orthocenter is point $$ H $$.
- **Altitudes**: $$ \overline{AH} $$, $$ \overline{BI} $$, $$ \overline{CJ} $$ - **Medians**: $$ \overline{AE} $$, $$ \overline{BF} $$, $$ \overline{CG} $$ - **Centroid**: Point $$ G $$ - **Orthocenter**: Point $$ H $$

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