The coordinates of a triangle ABC are A(1, 5), B(-2, 3), and C(0,-4); find the equation of the median AD?a)7x - 3y +-8=0b)5x - 4y + 15=0c)x +3y-16=0d)11x-4y+9=0Correct answer is option 'D'. Can you explain this answer?

Question

Answer

since AD is the median to BC, D will be mid point of BC. So coordinates of D

Equation of line passing through points A(1,5) and D(−1,−1/2) will be:

Hence, the correct option is (d).

Answer

Equation of Median AD:

- To find the equation of the median AD, we first need to determine the midpoint of side BC (point D) and then calculate the slope of AD.

Midpoint Calculation:
- The coordinates of B(-2, 3) and C(0, -4) are given.
- Using the midpoint formula:
- x-coordinate of D = (x-coordinate of B + x-coordinate of C) / 2 = (-2 + 0) / 2 = -1
- y-coordinate of D = (y-coordinate of B + y-coordinate of C) / 2 = (3 + (-4)) / 2 = -1/2
- Therefore, the midpoint D is (-1, -1/2).

Slope Calculation:
- The slope of AD can be determined using the coordinates of A(1, 5) and D(-1, -1/2).
- Slope (m) = (y2 - y1) / (x2 - x1) = (-1/2 - 5) / (-1 - 1) = -11/2

Equation of Median AD:
- Using the slope-intercept form (y = mx + c), we substitute the slope (m) and the midpoint (D) to find the equation of the median AD.
- y = (-11/2)x + c
- Substituting the coordinates of point D(-1, -1/2):
- -1/2 = (-11/2)(-1) + c
- -1/2 = 11/2 + c
- c = -11/2 - 1/2
- c = -6
- Therefore, the equation of the median AD is:
- y = (-11/2)x - 6
- Simplifying, we get: 11x - 4y + 9 = 0

Therefore, the correct answer is option 'D' (11x - 4y + 9 = 0).

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