Final answer:
To find a number t such that the line containing the points (t, -7) and (-8, 10) is perpendicular to the line containing the points (4, 7) and (1, 11), we can use the fact that the product of the slopes of two perpendicular lines is -1. The slope of the line passing through (4, 7) and (1, 11) is -4/3. Therefore, the solution for t is 10.
Step-by-step explanation:
To find a number t such that the line containing the points (t, -7) and (-8, 10) is perpendicular to the line containing the points (4, 7) and (1, 11), we can use the fact that the product of the slopes of two perpendicular lines is -1. The slope of the line passing through (4, 7) and (1, 11) can be found using the formula: m = (y2-y1)/(x2-x1). Using the given points, we have:
m = (11-7)/(1-4) = -4/3.
Since the slopes of the two lines are negative reciprocals, the slope of the line passing through (t, -7) and (-8, 10) is the negative reciprocal of m. Therefore, -4/3 * 1/m = -1. Solving for m:
m = -3/4
The slope of this line is -3/4. Using the formula for the slope between two points, we have: (10 - (-7))/(-8 - t) = -3/4.
Simplifying and solving for t, we get:
(10 + 7)/(-8 - t) = -3/4
17/(-8 - t) = -3/4
t = 10