The coordinates of the vertices of a triangle are \(A(0,0),\ B(8,0),\) and \(C(0,6)\). What is the location of the orthocenter?
\((0,0)\)
The vertices are \(A(0,0)\), \(B(8,0)\) and \(C(0,6)\). Side \(AB\) lies along the x-axis (both points have \(y=0\)) and side \(AC\) lies along the y-axis (both points have \(x=0\)).
The x-axis and y-axis meet at right angles, so \(AB\perp AC\). The triangle is therefore right-angled, with the right angle at vertex \(A(0,0)\).
The orthocentre is the common point of the three altitudes, where each altitude is the line drawn from a vertex perpendicular to the opposite side.
Consider the altitude from \(C\) to side \(AB\). Since \(AC\) is already perpendicular to \(AB\), the leg \(CA\) itself is that altitude, and it passes through \(A\).
Likewise the altitude from \(B\) to side \(AC\) is the leg \(BA\), which also passes through \(A\). Two altitudes already meet at \(A\), so the third must pass through \(A\) as well.
This illustrates the general property: in any right-angled triangle the orthocentre coincides with the vertex holding the right angle.
Hence the orthocentre is at \((0,0)\).
Similar Questions
Which of the following equations represents a line parallel to \(3x - 2y = 6\)?
What is the x-intercept of the line given by \(2x-3y=12\)?
Find the equation of a line that is parallel to \(y=3x-5\) and passes through the point \((1,2)\).
A drone flies along \(y = -x + 10\). Another follows \(y = x - 4\). At what point do they intersect?
A land plot is defined by points A(0,0), B(4,0), C(5,3), D(1,3). What is the area of the parallelogram representing this plot?
The equation of a line is \(2x-3y=6\). What is the slope of the line?
Find the equation of the line with slope -2 passing through the point (3, 5).
A triangle has vertices with coordinates A(1,2), B(5,2), and C(3,8). What is the y-coordinate of its centroid?
A line passes through the point (5, 7) and has a slope of -1. What is the equation of the line?
Important Questions from Co-ordinate Geometry
In which ratio the point (-3, p) divides the line segment joining the points (-5, -4) and (-2, 3)?
The area (in sq. units) of the triangle formed by the graphs of 8x + 3y = 24, 2x + 8 = y and the x-axis is:
In which quadrant both abscissa and ordinate are negative?
Find the slope of the line joining the points (3, -4) and (5, 2).
Find the value of K for which equation x – Ky = 2, 3x + 2y = 5 has unique solution.
