Question

Orthocentre of triangle having vertices as A (1,2), B(3,-4), C(0,6) is

• A. (-129, -37)
• B. (9, -1)
• C. (7, -3)
• D. (28, -16)

Answer 1

The Correct Option is A

To find the orthocentre of a triangle with given vertices, we must use the concept of altitudes. The orthocentre is the point where all three altitudes of the triangle intersect. Let's calculate it step by step for the given vertices of triangle \( A(1,2) \), \( B(3,-4) \), and \( C(0,6) \).

Find the slope of the side \( AB \):

The slope of line \( AB \) is \(\frac{{-4 - 2}}{{3 - 1}} = \frac{{-6}}{{2}} = -3\).

The slope of the altitude from point \( C \), which is perpendicular to \( AB \), is the negative reciprocal of the slope of \( AB \):

Hence, the slope of the altitude from \( C \) is \(\frac{1}{3}\).

Equation of the altitude from \( C \):

Using point-slope form, the equation of the altitude from \( C(0,6) \) is:

\( y - 6 = \frac{1}{3}(x - 0) \)

Simplifying this gives \( y = \frac{1}{3}x + 6 \).

Find the slope of the side \( BC \):

The slope of line \( BC \) is \(\frac{{6 - (-4)}}{{0 - 3}} = \frac{10}{-3} = -\frac{10}{3}\).

The slope of the altitude from point \( A \), which is perpendicular to \( BC \), is the negative reciprocal of the slope of \( BC \):

Hence, the slope of the altitude from \( A \) is \(\frac{3}{10}\).

Equation of the altitude from \( A \):

Using point-slope form, the equation of the altitude from \( A(1,2) \) is:

\( y - 2 = \frac{3}{10}(x - 1) \)

Simplifying this gives \( y = \frac{3}{10}x + \frac{17}{10} \).

Finding the intersection of the two altitudes:

The orthocentre is the intersection of these two altitudes. Set the equations equal to each other:

\( \frac{1}{3}x + 6 = \frac{3}{10}x + \frac{17}{10} \)

Clear the fractions by multiplying through by 30:

\( 10x + 180 = 9x + 51 \)

Solving gives \( x = -129 \).

Substitute \( x = -129 \) back into one of the altitude equations, \( y = \frac{1}{3}x + 6 \):

\( y = \frac{1}{3}(-129) + 6 = -43 + 6 = -37 \).

Thus, the orthocentre of the triangle with vertices \( A(1,2) \), \( B(3,-4) \), \( C(0,6) \) is (-129, -37), which matches the correct answer given in options.