Answer: The distance between points A and B is 32 units.
Step-by-step explanation:
The general equation for a line in slope-intercept form is:
y = a*x +b
Where a is the slope, and b is the y-intercept.
We know that two lines are parallel if the lines have the same slope and different y-intercept.
Now, in this case we have the line:
5*x + 4*y = 16
We can rewrite this in slope-intecept form if we isolate the "y" in the left side:
4*y = -5*x + 16
y = (-5/4)*x + 16/4
y = (-5/4)*x+ 4
The slope is (-5/4) and the y-intercept is 4.
We know that line L₂ is parallel to this line, then this line will also have a slope equal to (-5/4) and a y-intercept equal to c.
y = (-5/4)*x + c
And we know that this line passes through the point (8, 15)
This means that when x = 8, the value of y must be 15.
We could just replace these two values in the above equation to find the value of c.
15 = (-5/4)*8 + c
15 + (5/4)*8 = c = 25
Then the line L₂ is:
y = (-5/4)*x + 25.
Now, we know that this line passes the x-axis at the point A.
The line will pass through the x-axis when y = 0, then we need to find the value of x such that y = 0.
0 = (-5/4)*x + 25
(5/4)*x = 25
x = 25/(5/4) = 20
Then point A is the point (20, 0)
And point B is when the line passes through the y-axis, this is when x = 0.
y = (-5/4)*0 + 25
y = 25
Then point B is the point (0, 25)
Now we want to find the distance between points A and B, which is equal to the distance between points (20, 0) and (0, 25).
When we have two points (a, b) and (c, d), the distance between them is:
distance = √( (a - c)^2 + (b - d)^2)
In this case, the distance between (20, 0) and (0, 25) is:
distance = √( (20 - 0)^2 + (0 -25)^2) = 32
The distance between points A and B is 32 units.
