Question

Find all values of x so that the triangle with vertices A=(x,−3,−5),B=(−3,−5,−5), and C=(−2,0,−8) has a right angle at A Enter your answer as a comma-separated list of values Use the square root symbol ' ✓ ' where needed to give an exact value for your answer. x=

Answer 1

To find the values of x for which the triangle has a right angle at A, we can use the concept of the dot product. Two vectors are perpendicular if and only if their dot product is zero. We find x = 1 or x = 5 as the values that satisfy the condition.

Step-by-step explanation:

To find the values of x for which the triangle with vertices A=(x,-3,-5), B=(-3,-5,-5), and C=(-2,0,-8) has a right angle at A, we can use the concept of the dot product. Two vectors are perpendicular if and only if their dot product is zero. Let's find the vectors AB and AC and check if their dot product is zero.

1. Vector AB: AB = B - A = (-3,-5,-5) - (x,-3,-5) = (-3-x, -5+3, -5+5) = (-3-x,-2,0)
2. Vector AC: AC = C - A = (-2,0,-8) - (x,-3,-5) = (-2-x, 0+3, -8+5) = (-2-x,3,-3)
3. Dot product: AB · AC = (-3-x)(-2-x) + (-2)(3) + (0)(-3) = 6 + 5 + 0 = 11 - 6x + x^2 - 6 = x^2 - 6x + 5

We want the dot product to be zero, so we solve the equation x^2 - 6x + 5 = 0. Factoring or using the quadratic formula, we find that x = 1 or x = 5 are the values that satisfy the condition. Therefore, the triangle has a right angle at A when x is either 1 or 5.