If AD¯¯¯¯¯¯¯¯ is the altitude to BC¯¯¯¯¯¯¯¯, what is the slope of AD¯¯¯¯¯¯¯¯?
A=(0,1) B=(7,-2) C=(3,8)
A. 5/2
B. -2/5
C. -5/2
D. 2/5
The slope of BC is -5/2
AD⊥BC, so ...
To find the slope of AD¯¯¯¯¯¯¯¯, we need to calculate the slope between points A(0,1) and D(x,y), where D lies on the line AD¯¯¯¯¯¯¯¯ and is perpendicular to BC¯¯¯¯¯¯¯¯ (which means AD¯¯¯¯¯¯¯¯ is the altitude to BC¯¯¯¯¯¯¯¯).
The slope between two points (x₁, y₁) and (x₂, y₂) can be calculated using the formula:
slope = (y₂ - y₁) / (x₂ - x₁)
In this case, we already know the coordinates of point A as (0,1). To find the coordinates of point D, we need to determine the equation of line BC¯¯¯¯¯¯¯¯ using points B(7,-2) and C(3,8).
Step 1: Finding the equation of line BC¯¯¯¯¯¯¯¯
First, find the slope of BC¯¯¯¯¯¯¯¯, which is the slope between points B and C:
slope_BC = (y_C - y_B) / (x_C - x_B)
= (8 - (-2)) / (3 - 7)
= 10 / (-4)
= -5/2
Next, use the slope-intercept form of a line (y = mx + b) to find the equation of line BC¯¯¯¯¯¯¯¯:
Using point B (7,-2),
-2 = (-5/2)(7) + b
-2 = -35/2 + b
b = -2 + 35/2
b = -4/2 + 35/2
b = 31/2
So, the equation of line BC¯¯¯¯¯¯¯¯ is:
y = (-5/2)x + 31/2
Step 2: Finding the equation of line AD¯¯¯¯¯¯¯¯
Since AD¯¯¯¯¯¯¯¯ is perpendicular to BC¯¯¯¯¯¯¯¯, its slope will be the negative reciprocal of the slope of BC¯¯¯¯¯¯¯¯. So, the slope of AD¯¯¯¯¯¯¯¯ will be:
slope_AD = -1 / slope_BC
= -1 / (-5/2)
= 2/5
Therefore, the slope of AD¯¯¯¯¯¯¯¯ is 2/5. So, the correct answer is:
D. 2/5