points A(-1,-2), B(5,10),C(0,5). How can I deduce that the perpendicular distance from the point C to the line AB is root(5) ?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find the equation of the line AB such that:

`y - y_A = m_(AB)(x - x_A)`

`m_(AB)`  denotes the slope of the line AB.

`m_(AB) = (y_B - y_A)/(x_B - x_A) =gtm_(AB) = (10+2)/(5+1) =gt m_(AB) = 12/6 =gt m_(AB) = 2` 

y + 2 = 2(x + 1)

You need to open the brackets and to move the terms to the left side:

y + 2 - 2x - 2 = 0 => y - 2x = 0

You need to remember the formula of distance from point M`(x_M,y_M) ` to the line ax+by+c = 0.

`d = |ax_M + by_M + c|/sqrt(a^2 + b^2)`

Hence, the distance from C(0,5) to the line AB is:

`d = |-2*0 + 1*5 + 0|/sqrt((-2)^2 + 1^2)`

`` d = `5/sqrt5 ` => d =`5sqrt5/5`  => d = `sqrt 5`

Hence, the distance from C(0,5) to the line AB is d=`sqrt 5` .

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