A triangle has one side that lies along the line y=1/4x and another that lies along the line y=-1/4x. Which of the following points could be a vertex of the triangle?
A. (4,4)
B. (0,0)
C. (-4,-2)
D. (1,8)
E. (-2,-8)
clearly the lines intersect where
1/4 x = -1/4 x
1/2 x = 0
x = 0
so, y=0
Looks like B to me.
You can easily check to see that none of the other points lies on either line.
To determine which points could be vertices of the triangle, we need to find the third vertex that completes the triangle.
One side of the triangle lies along the line y = 1/4x, and another side lies along the line y = -1/4x.
Let's start by writing the equations of these lines in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
For the first line, y = 1/4x, the slope is 1/4 and the y-intercept is 0. This means the line goes through the origin (0,0) and has a positive slope.
For the second line, y = -1/4x, the slope is -1/4 and the y-intercept is 0. This means the line goes through the origin (0,0) and has a negative slope.
Both lines pass through the origin (0,0), so the origin (0,0) is definitely a vertex of the triangle.
To find the third vertex, we need to identify a point that lies on only one of the lines.
Let's check each option:
A. (4,4)
Substituting the x and y coordinates into the equations of the lines:
For y = 1/4x: 4 = 1/4(4) = 1, not true
For y = -1/4x: 4 = -1/4(4) = -1, not true
Since the point (4,4) does not lie on either line, it cannot be a vertex of the triangle.
B. (0,0)
We already determined that (0,0) is a vertex of the triangle.
C. (-4,-2)
Substituting the x and y coordinates into the equations of the lines:
For y = 1/4x: -2 = 1/4(-4) = -1, not true
For y = -1/4x: -2 = -1/4(-4) = 1, not true
Since the point (-4,-2) does not lie on either line, it cannot be a vertex of the triangle.
D. (1,8)
Substituting the x and y coordinates into the equations of the lines:
For y = 1/4x: 8 = 1/4(1) = 1/4, not true
For y = -1/4x: 8 = -1/4(1) = -1/4, not true
Since the point (1,8) does not lie on either line, it cannot be a vertex of the triangle.
E. (-2,-8)
Substituting the x and y coordinates into the equations of the lines:
For y = 1/4x: -8 = 1/4(-2) = -1/2, not true
For y = -1/4x: -8 = -1/4(-2) = 1/2, not true
Since the point (-2,-8) does not lie on either line, it cannot be a vertex of the triangle.
Therefore, the only possible vertex of the triangle is B. (0,0).