Question

If a triangle \(ABC\) has vertices \(A(1,-6),B(2,-3)\) and \(C(3,-2)\), then the coordinates of its orthocenter are....

• A. \((-6,1)\)
• B. \((6,1)\)
• C. \((-2,3)\)
• D. \((3,2)\)

Hint:

Find two altitudes using perpendicular slopes and intersect them.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The orthocenter is the point where the three altitudes of a triangle meet. An altitude from a vertex is perpendicular to the opposite side. Two altitudes are enough to find the point.

Step 2: Altitude from A:
Slope of \(BC\) is \(\dfrac{-2-(-3)}{3-2}=1\). So the altitude from \(A\) has slope \(-1\) and passes through \(A(1,-6)\):
\[ y+6=-(x-1)\ \Rightarrow\ y=-x-5 \]

Step 3: Altitude from B:
Slope of \(AC\) is \(\dfrac{-2+6}{3-1}=2\). So the altitude from \(B\) has slope \(-\tfrac12\) and passes through \(B(2,-3)\):
\[ y+3=-\tfrac12(x-2)\ \Rightarrow\ y=-\tfrac{x}{2}-2 \]

Step 4: Intersect:
Equate: \(-x-5=-\tfrac{x}{2}-2\), so \(-\tfrac{x}{2}=3\) and \(x=-6\). Then \(y=-(-6)-5=1\).

Step 5: Choose:
The orthocenter is \((-6,1)\), option (A). Option (B) swaps the sign of \(x\) and option (C) and (D) do not satisfy the altitude from \(A\).

Final Answer:
The orthocenter is (-6, 1). \[ \boxed{(-6,\,1)} \]