The condition for a right angle at vertex A is that the dot product of the vectors AB and AC be zero.
So we first need to find the vectors AB and AC:
AB = B - A = (-6,-10,-7) - (-9,-7,2) = (3,-3,-9)
AC = C - A = (-18,-15,x) - (-9,-7,2) = (-9,-8,x-2)
Now we calculate the dot product of AB and AC:
AB · AC = (3,-3,-9) · (-9,-8,x-2) = -27 + 24 - 9(x-2) = -3x + 63
For a right angle at vertex A, AB · AC must be zero:
-3x + 63 = 0
Solving for x, we get:
x = 21
Therefore, the value of x that makes the triangle have a right angle at vertex A is x = 21.
Find x so that the triangle with vertices A=(-9, -7,2) B=(-6,-10,-7) and C=(-18,-15,x) has a right angle at A.
2 answers
The bot was actually correct up to :
AB · AC = (3,-3,-9) · (-9,-8,x-2) = -27 + 24 - 9(x-2)
but then it messed up
should be
-27 + 24 - 9x + 18 = 0
-9x = -15
x = 15/9 = 5/3
AB · AC = (3,-3,-9) · (-9,-8,x-2) = -27 + 24 - 9(x-2)
but then it messed up
should be
-27 + 24 - 9x + 18 = 0
-9x = -15
x = 15/9 = 5/3