Problem 3 Finding the Orthocenter

Δ A B C cap delta eh b c  has vertices A(1, 3), B(2, 7), and C(6, 3). What are the coordinates of the orthocenter of Δ A B C ? cap delta eh b c question mark

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• Step 1 Find the equation of the line containing the altitude to A C ¯ . eh c bar , .  Since A C ¯ eh c bar  is horizontal, the line containing the altitude to A C ¯ eh c bar  is vertical. The line passes through the vertex B(2, 7). The equation of the line is x = 2 . x equals 2 , .

• Step 2 Find the equation of the line containing the altitude to B C ¯ . b c bar , .  The slope of the line containing B C ¯ b c bar  is 3 − 7 6 − 2 = − 1 . fraction 3 minus 7 , over 6 minus 2 end fraction . equals negative 1 .  Since the product of the slopes of two perpendicular lines is − 1 , negative 1 comma  the line containing the altitude to B C ¯ b c bar  has slope 1.

  The line passes through the vertex A(1, 3). The equation of the line is y − 3 = 1 ( x − 1 ) , y minus , 3 equals 1 , open x minus 1 close comma  which simplifies to y = x + 2 . y equals . x plus 2 , .

• Step 3 Find the orthocenter by solving this system of equations: x = 2 y = x + 2 table with 2 rows and 1 column , row1 column 1 , x equals 2 , row2 column 1 , y equals x plus 2 , end table

  y = 2 + 2 Substitute  2  for  x  in the second equation . y = 4 Simplify . table with 2 rows and 2 columns , row1 column 1 , y equals 2 plus 2 , column 2 cap substitute . 2 , for , x . inthesecondequation . . , row2 column 1 , y equals 4 , column 2 cap simplify , . , end table

  The coordinates of the orthocenter are (2, 4).

✓ Got It?

3. Δ D E F cap delta d e f  has vertices D(1, 2), E(1, 6), and F(4, 2). What are the coordinates of the orthocenter of Δ D E F ? cap delta d e f question mark