The angular points of a triangle are A( - 1, - 7), B(5,1), C(1,4).The equation of bisector of angle ∠ABC is

A. x=7y+2 B. 7y=x+2 C. y = 7x + 2 D. 7x=y+2

To find the equation of the angle bisector of ∠ABC, we need to use the formula for the angle bisector in a triangle. We are given the points A(-1, -7), B(5, 1), and C(1, 4).

Step 1: Find the slopes of lines AB and BC
The slope of line AB is given by:

m_AB = (y2 - y1) / (x2 - x1) = (1 - (-7)) / (5 - (-1)) = 8 / 6 = 4 / 3

The slope of line BC is:

m_BC = (y2 - y1) / (x2 - x1) = (4 - 1) / (1 - 5) = 3 / (-4) = -3 / 4

Step 2: Use the formula for the angle bisector
The angle bisector of ∠ABC divides the angle between AB and BC in the ratio of the lengths of the opposite sides. The formula for the slope of the angle bisector of ∠ABC is:

m = (m_AB * m_BC - 1) / (m_AB + m_BC)

Substitute the values of m_AB and m_BC into the formula:

m = [(4 / 3) * (-3 / 4) - 1] / [(4 / 3) + (-3 / 4)] = [(-12 / 12) - 1] / [(16 / 12) - (9 / 12)] = [-1 - 1] / [7 / 12] = -2 / (7 / 12) = -2 * (12 / 7) = -24 / 7

So, the slope of the angle bisector is -24/7.

Step 3: Find the equation of the angle bisector
The equation of a line with slope m passing through a point (x1, y1) is:

y - y1 = m(x - x1)

Here, we have point B(5, 1) and slope m = -24/7. Substitute these into the equation:

y - 1 = (-24 / 7)(x - 5)

Multiply out:

y - 1 = (-24 / 7)x + (24 * 5 / 7) y - 1 = (-24 / 7)x + 120 / 7

Add 1 to both sides:

y = (-24 / 7)x + 120 / 7 + 7 / 7 y = (-24 / 7)x + 127 / 7

Multiply both sides by 7 to eliminate the denominator:

7y = -24x + 127

Rearrange to match the given options:

7y = -24x + 127

Now, we compare this with the options given. The closest match is option D, 7x = y + 2.

Thus, the correct answer is:

D. 7x = y + 2.