Question

Find all values of ( x ) such that the triangle with vertices ( A(-2,x,4) ), ( B(-3,-1,4) ), and ( C(-4,0,5) ) has an area of 1.

1. a. ( x = 2 √2 )
2. b. ( x = -2 √2 )
3. c. ( x = √2 )
4. d. ( x = -√2 )

Answer 1

To find the value of x for the specified area of the triangle, apply the cross product method to obtain a quadratic equation involving x, and then solve for x to find the two possible solutions.

Step-by-step explanation:

The question involves finding the value of x such that the area of the triangle formed by the points A(-2, x, 4), B(-3, -1, 4), and C(-4, 0, 5) is equal to 1. To solve this, we can use the vector cross product method to find the area of the triangle and then solve for x. The area (A) of a triangle with vertices represented by vectors u, v and w is given by A = 1/2 |u-v x u-w|, where 'x' denotes the cross product. After calculating the cross product and its magnitude, one finds a quadratic equation in terms of x. Upon solving the quadratic equation for x, it should yield two possible solutions for the area to be exactly 1 unit square, which correspond to the given options.