Hi please help :) Find the point of intersection of the line AB (3x + 4y - 16 = 0) by the perpendicular line L which passes through point P (7,5).

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hala718 | High School Teacher | (Level 1) Educator Emeritus

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Given the equation of the line AB :

3x+4y -16 = 0

We will rewrite into the slope format:

==> 4y = -3x +16

==> y= (-3/4)x + 4

Then we notice that the slope is -3/4

Then, the slope of the perpendicular line is 4/3.

Now we will find  the equation of the perpendicular line L.

Given the point P(7,5) and the slope is 4/3.

==> y-y1= m (x-x1).

==> y-5 = (4/3) (x-7)

==> y= (4/3)x - 28/3 + 5

==> y= (4/3)x  -13/3

Now we have the equation of the line AB and the line L.

Now we will find the intersection point.

==> (-3/4)x + 4 = (4/3)x - 13/3

==> (4/3)x + (3/4)x = 4+ 13/3

==> (16+9)x/12 =  25/3

==> 25x/12 = 25/3

==> x/4 = 1

==> x = 4

==> y= (-3/4)x + 4 = -3 + 4 = 1

Then, the point of intersection is (4,1).

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