What is the orthocenter of a triangle with corners at #(1 ,3 )#, #(6 ,9 )#, and (2 ,4 )#?
The orthocenter of triangle is
Let
Let
Let
Slope of
Slope of
Taking
From equn.
Hence, the orthocenter of triangle is
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To find the orthocenter of a triangle, we need to first determine the altitudes of the triangle, which are perpendicular lines drawn from each vertex to the opposite side. Then, the point where these altitudes intersect is the orthocenter.
Let's denote the vertices of the triangle as ( A(1, 3) ), ( B(6, 9) ), and ( C(2, 4) ).
We can start by finding the equations of the lines passing through each vertex and perpendicular to the opposite side. The equation of a line passing through a point ( (x_1, y_1) ) with slope ( m ) is given by ( y  y_1 = m(x  x_1) ). For a line perpendicular to a given line with slope ( m ), the slope of the perpendicular line is the negative reciprocal of ( m ).

For the line through ( A ) perpendicular to ( BC ) (the line passing through ( B ) and ( C )):
 Slope of ( BC = \frac{94}{62} = \frac{5}{4} )
 Slope of the line perpendicular to ( BC ) passing through ( A ): ( \frac{4}{5} )
 Equation of the line: ( y  3 = \frac{4}{5}(x  1) )

For the line through ( B ) perpendicular to ( AC ) (the line passing through ( A ) and ( C )):
 Slope of ( AC = \frac{43}{21} = 1 )
 Slope of the line perpendicular to ( AC ) passing through ( B ): ( 1 )
 Equation of the line: ( y  9 = 1(x  6) )

For the line through ( C ) perpendicular to ( AB ) (the line passing through ( A ) and ( B )):
 Slope of ( AB = \frac{93}{61} = \frac{6}{5} )
 Slope of the line perpendicular to ( AB ) passing through ( C ): ( \frac{5}{6} )
 Equation of the line: ( y  4 = \frac{5}{6}(x  2) )
Next, we find the intersection point of these three lines, which will be the orthocenter of the triangle. Solving these equations simultaneously will give us the coordinates of the orthocenter.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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