Math Problem Statement

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

Solution

To find the reflection of the orthocenter of triangle OABOAB with respect to the line y=xy = x, we'll first need to determine the orthocenter OO of triangle ABCABC. Then, we reflect this orthocenter across the line y=xy = x.

Step 1: Find the orthocenter of triangle ABCABC.

Given the vertices:

  • A(1,2)A(1, 2)
  • B(3,4)B(3, 4)
  • C(5,3)C(-5, 3)

The slopes of the sides opposite to each vertex are:

  • Slope of BCBC: mBC=433(5)=18m_{BC} = \frac{4 - 3}{3 - (-5)} = \frac{1}{8}
  • Slope of CACA: mCA=231(5)=16m_{CA} = \frac{2 - 3}{1 - (-5)} = \frac{-1}{6}
  • Slope of ABAB: mAB=4231=1m_{AB} = \frac{4 - 2}{3 - 1} = 1

Now, we find the equations of the altitudes:

  • The slope of the altitude from AA is perpendicular to BCBC, so it will have a slope of m=8m = -8. Its equation using point-slope form is: y2=8(x1)y - 2 = -8(x - 1) y=8x+10y = -8x + 10
  • The slope of the altitude from BB is perpendicular to CACA, so it will have a slope of m=6m = 6. Its equation using point-slope form is: y4=6(x3)y - 4 = 6(x - 3) y=6x14y = 6x - 14

To find the orthocenter OO, solve the system of equations: 8x+10=6x14-8x + 10 = 6x - 14 14x=24x=127-14x = -24 \quad \Rightarrow \quad x = \frac{12}{7} Substituting x=127x = \frac{12}{7} into one of the altitude equations: y=8(127)+10=967+707=267y = -8\left(\frac{12}{7}\right) + 10 = \frac{-96}{7} + \frac{70}{7} = \frac{-26}{7} Thus, the orthocenter OO is (127,267)\left(\frac{12}{7}, \frac{-26}{7}\right).

Step 2: Reflect the orthocenter OO across the line y=xy = x.

To reflect a point (x,y)(x, y) across the line y=xy = x, swap the coordinates. The reflection of OO is: O=(267,127)O' = \left(\frac{-26}{7}, \frac{12}{7}\right)

Final Answer:

The reflection of the orthocenter of triangle OABOAB with respect to the line y=xy = x is (267,127)\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.