Answer:
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
