if vertices of a triangle ABC are A(1,2) B( 3,4) and C(-5,3) and its orthocentre is O, then the reflection of orthocentre of triangle OAB with respect to line y=x will be

To find the reflection of the orthocenter of triangle \( OAB \) with respect to the line \( y = x \), we'll first need to determine the orthocenter \( O \) of triangle \( ABC \). Then, we reflect this orthocenter across the line \( y = x \).

### Step 1: Find the orthocenter of triangle \( ABC \).

Given the vertices:
- \( A(1, 2) \)
- \( B(3, 4) \)
- \( C(-5, 3) \)

The slopes of the sides opposite to each vertex are:
- Slope of \( BC \): \[ m_{BC} = \frac{4 - 3}{3 - (-5)} = \frac{1}{8} \]
- Slope of \( CA \): \[ m_{CA} = \frac{2 - 3}{1 - (-5)} = \frac{-1}{6} \]
- Slope of \( AB \): \[ m_{AB} = \frac{4 - 2}{3 - 1} = 1 \]

Now, we find the equations of the altitudes:
- The slope of the altitude from \( A \) is perpendicular to \( BC \), so it will have a slope of \( m = -8 \). Its equation using point-slope form is:
  \[
  y - 2 = -8(x - 1)
  \]
  \[
  y = -8x + 10
  \]
- The slope of the altitude from \( B \) is perpendicular to \( CA \), so it will have a slope of \( m = 6 \). Its equation using point-slope form is:
  \[
  y - 4 = 6(x - 3)
  \]
  \[
  y = 6x - 14
  \]

To find the orthocenter \( O \), solve the system of equations:
\[
-8x + 10 = 6x - 14
\]
\[
-14x = -24 \quad \Rightarrow \quad x = \frac{12}{7}
\]
Substituting \( x = \frac{12}{7} \) into one of the altitude equations:
\[
y = -8\left(\frac{12}{7}\right) + 10 = \frac{-96}{7} + \frac{70}{7} = \frac{-26}{7}
\]
Thus, the orthocenter \( O \) is \( \left(\frac{12}{7}, \frac{-26}{7}\right) \).

### Step 2: Reflect the orthocenter \( O \) 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 \( OAB \) 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.

Explore how to find the reflection of the orthocenter of triangle OAB with respect to the line y = x using coordinate geometry. Step-by-step solution provided for a triangle with vertices A(1, 2), B(3, 4), and C(-5, 3), and the orthocenter O at (12/7, -26/7).

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