The vertex A of a triangle ABC is the point (-2, 3) whereas the line perpendicular to the sides AB and AC are x – y – 4 = 0 and 2x – y – 5 = 0 respectively. The right bisectors of sides meet at P(3/2 , 5/2) . Then the equation of the median of side BC isa)5x + 2y = 10b)5x – 2y = 16c)2x – 5y = 10d)none of theseCorrect answer is option 'D'. Can you explain this answer?

Question

The vertex A of a triangle ABC is the point (-2, 3) whereas the line perpendicular to the sides AB and AC are x &ndash; y &ndash; 4 = 0 and 2x &ndash; y &ndash; 5 = 0 respectively. The right bisectors of sides meet at P(3/2 , 5/2) . Then the equation of the median of side BC isa)5x + 2y = 10b)5x &ndash; 2y = 16c)2x &ndash; 5y = 10d)none of theseCorrect answer is option 'D'. Can you explain this answer?

Answer

To find the equation of the line perpendicular to sides AB and AC passing through vertex A, we need to first find the slopes of sides AB and AC.

The slope of side AB can be found using the formula: slope = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points on the line.

Given that point A is (-2, 3) and point B is unknown, we cannot directly find the slope of AB. However, we can find the slope of the perpendicular line, as the product of the slopes of perpendicular lines is -1.

So, let's say the slope of AB is m. The slope of the perpendicular line will then be -1/m.

Similarly, let's find the slope of AC. Given that point A is (-2, 3) and point C is unknown, we cannot directly find the slope of AC. However, we can find the slope of the perpendicular line, which will be -1/m' (where m' is the slope of AC).

Now, we can find the equations of the two perpendicular lines:

1. Perpendicular to AB:
Using point-slope form, the equation of the line passing through A with slope -1/m is:
y - 3 = (-1/m)(x + 2)

2. Perpendicular to AC:
Using point-slope form, the equation of the line passing through A with slope -1/m' is:
y - 3 = (-1/m')(x + 2)

Note: Without the coordinates of points B and C, we cannot determine the exact equations of these perpendicular lines. We can only express them in terms of their slopes, m and m'.

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