Problem 19 - College Algebra and Calculus: An Applied Approach

Question

Find the coordinates of the midpoint of the line segment joining the two points. $$ (-5,-2,5),(6,3,-7) $$

Step-by-Step Solution

Answer

The coordinates of the midpoint are (0.5, 0.5, -1)

Step 1: Identify the coordinates

We have the points \( A(-5, -2, 5) \) and \( B(6, 3, -7) \). For point A, \( x1 = -5\), \(y1 = -2\) and \(z1 = 5\), and for point B, \( x2 = 6 \), \( y2 = 3 \) and \( z2 = -7 \). We can now substitute these into the formula to find the midpoint.

Step 2: Substitute into the formula

The formula for the midpoint in 3D is \( M = \left( \frac{x1 + x2}{2}, \frac{y1 + y2}{2}, \frac{z1 + z2}{2} \right) \).\n We substitute the given coordinates into the formula, we get \( M = \left( \frac{-5+6}{2}, \frac{-2+3}{2}, \frac{5-7}{2} \right)\)

Step 3: Simplify the expressions

Simplifying each expression in the coordinates, we get \( M = \left( \frac{1}{2}, \frac{1}{2}, \frac{-2}{2} \right) = (0.5, 0.5, -1)\). So, the coordinates of the midpoint of the line segment AB are \( (0.5, 0.5, -1) \)