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jolli1 [7]
3 years ago
5

Line l contains the points (1,5) and (4, -4). point P has the coordinates (-1, 1)

Mathematics
1 answer:
Colt1911 [192]3 years ago
3 0

Answer:

The distance from p to l is \sqrt{10}\ units

Step-by-step explanation:

we know that

The distance between point p from line l is equal to the perpendicular segment from line l to point p

step 1

<em>Find the slope of line l</em>

we have the points

(1,5) and (4, -4)

The formula to calculate the slope between two points is equal to

m=\frac{y2-y1}{x2-x1}

substitute the values

m=\frac{-4-5}{4-1}

m=\frac{-9}{3}

m=-3

step 2

Find the equation of the line l

The equation in point slope form is equal to

y-y1=m(x-x1)

we have

m=-3

point\ (1,5)

substitute

y-5=-3(x-1) -----> equation A

step 3

Find the slope of the line perpendicular to the line l

Remember that

If two lines are perpendicular, then their slopes are opposite reciprocal (The product of their slopes is equal to -1)

m_1*m_2=-1

we have

m_1=-3 ---> slope of line l

therefore

m_2=\frac{1}{3} ----> slope of the line perpendicular to line l

step 4

Find the equation of the line perpendicular to line l that passes through the point p

The equation in point slope form is equal to

y-y1=m(x-x1)

we have

m=\frac{1}{3}

point\ p(-1,1)

substitute

y-1=\frac{1}{3}(x+1) -----> equation B

step 5

Solve the system of equations

y-5=-3(x-1) -----> equation A

y-1=\frac{1}{3}(x+1) -----> equation B

Solve the system by graphing

The solution of the system is the intersection point both graphs

The solution is the point q(2,2)

see the attached figure

step 6

we know that

The distance between the point p and the line l is equal to the distance between the point p and the point q

the formula to calculate the distance between two points is equal to

d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}

we have the points

p(-1,1) and q(2,2)

substitute the values

d_p_q=\sqrt{(2-1)^{2}+(2+1)^{2}}

d_p_q=\sqrt{(1)^{2}+(3)^{2}}

d_p_q=\sqrt{10}\ units

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