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

Question

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

Answer

Finding the Midpoint of BC
To find the equation of median AD in triangle ABC, we first need to determine the midpoint D of side BC.
- Coordinates of B: (-2, 3)
- Coordinates of C: (0, -4)
The midpoint D can be calculated as:
- D = ((x1 + x2)/2, (y1 + y2)/2)
- D = ((-2 + 0)/2, (3 + (-4))/2) = (-1, -0.5)
Finding the Slope of AD
Next, we calculate the slope of the line segment AD, where A is (1, 5) and D is (-1, -0.5).
- Slope of AD = (y2 - y1) / (x2 - x1)
- Slope of AD = (-0.5 - 5) / (-1 - 1) = -5.5 / -2 = 2.75
Equation of Line AD
Using point-slope form (y - y1 = m(x - x1)), where m is the slope and (x1, y1) is point A:
- y - 5 = 2.75(x - 1)
Rearranging this gives:
- y - 5 = 2.75x - 2.75
- y = 2.75x + 2.25
To convert this into standard form (Ax + By + C = 0):
- 2.75x - y + 2.25 = 0
- To eliminate decimals, multiply through by 4:
- 11x - 4y + 9 = 0
Conclusion
Hence, the equation of the median AD is:
- 11x - 4y + 9 = 0
This corresponds to option 'd', which appears to be mistakenly referenced as option 'c' in the original question.

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