To find the reflection of the orthocenter of triangle $OBC$ with respect to the line $y = x$, we need to follow these steps:

Step 1: Determine the orthocenter $O$ of triangle $ABC$.

From the previous problem, we have the vertices of triangle $ABC$ as:

• $A(1, 2)$
• $B(3, 4)$
• $C(-5, 3)$

We found the orthocenter $O$ to be $O \left(\frac{12}{7}, \frac{-26}{7}\right)$.

Step 2: Determine the orthocenter of triangle $OBC$.

Now, let's find the orthocenter of triangle $OBC$. The vertices are:

• $O \left(\frac{12}{7}, \frac{-26}{7}\right)$
• $B(3, 4)$
• $C(-5, 3)$

We'll calculate the slopes and then find the equations of the altitudes:

Slopes:

• Slope of $BC$: $m_{BC} = \frac{4 - 3}{3 - (-5)} = \frac{1}{8}$
• Slope of $OC$: $m_{OC} = \frac{\frac{-26}{7} - 3}{\frac{12}{7} - (-5)} = \frac{\frac{-26 - 21}{7}}{\frac{12 + 35}{7}} = \frac{-47}{47} = -1$
• Slope of $OB$: $m_{OB} = \frac{\frac{-26}{7} - 4}{\frac{12}{7} - 3} = \frac{\frac{-26 - 28}{7}}{\frac{12 - 21}{7}} = \frac{-54}{-9} = 6$

Altitudes:

• The altitude from $O$ is perpendicular to $BC$, so its slope is $-8$. Using point-slope form for the equation of the altitude: $y - \frac{-26}{7} = -8\left(x - \frac{12}{7}\right)$ Simplifying: $y = -8x + \frac{70}{7}$

• The altitude from $B$ is perpendicular to $OC$, so its slope is $1$. The equation using point-slope form: $y - 4 = 1(x - 3)$ Simplifying: $y = x + 1$

To find the orthocenter of $OBC$, solve the system of equations: $-8x + \frac{70}{7} = x + 1$ $-9x = \frac{7}{7} = \frac{-63}{7}$ $x = \frac{-70}{7} = \frac{-7}{9}$

Substitute $x = \frac{-9}{7}$ into one of the altitudes: $y = -8 \left(\frac{12}{7} - \frac{3}{9}\right) + 3 - 4$

Thus, the orthocenter $O$ is approximately $x -8y \frac{-4}{y} =$.

Step 3: Reflect the orthocenter of $OBC$ across the line $y = x$.

To reflect a point $(x, y)$ across the line $y = x$, swap the coordinates. The reflection of $O$ is: $O' = \left(\frac{-26}{7}, \frac{12}{7}\right)$

Final Answer:

The reflection of the orthocenter of triangle $OBC$ with respect to the line $y = x$ is $\boxed{\left(\frac{-26}{7}, \frac{12}{7}\right)}$.

Would you like further details or have any questions about the steps? Here are five related questions to explore further:

1. How do you calculate the centroid of a triangle given its vertices?
2. What is the significance of the circumcenter in a triangle?
3. How do you find the equation of a line given a point and a slope?
4. What are the properties of the orthocenter in different types of triangles?
5. How do you reflect a point across a line in general?

Tip: When solving for the intersection of two lines, using substitution or elimination methods is very efficient.