The coordinates of 3 points are A(1, 1), B(-1, 4) and C(6, k). Find the value of k if AB is perpendicular to BC

k = 13/2

Step-by-step explanation:

AB is perpendicular to BC means that, AB and BC form the opposite and adjacent sides of a triangle. While AC is the hypotenuse.

Length of AB is;

|AB|² = (4 - 1)² + (-1 - 1)²

|AB|² = 13

Similarly,

|BC|² = (k - 4)² + (6 + 1)²

|BC|² = k² - 8x + 16 + 49

|BC|² = k² - 8x + 65

Also;

|AC|² = (k - 1)² + (6 - 1)²

|AC|² = k² - 2k + 1 + 25

|AC|² = k² - 2k + 26

Thus,from pythagoras theorem;

|AC|² = |AB|² + |BC|²

k² - 2k + 26 = 13 + k² - 8x + 65

k² will cancel out and we rearrange to get;

8k - 2k = 65 - 26

6k = 39

k = 39/6

Divide both numerator and denominator by 3 to get;

k = 13/2

there are many solutions.

step-by-step explanation:

step 1: simplify both sides of the equation.

4x+6=4x^2+10x−4

step 2: subtract 4x^2+10x-4 from both sides.

4x+6−(4x^2+10x−4)=4x^2+10x−4−(4x^2+10x−4)

−4x^2−6x+10=0

step 3: factor left side of equation.

−2(2x+5)(x−1)=0

step 4: set factors equal to 0.

x=-5/2 or x=1