1. a triangle has vertices at (0,0), (2,1),(1,3). About which line could the triangle be resolved, in order to form one cone?
A) the line at x=2y
B) the line at y=3x
C) the x-axis
D) the y-axis
Could someone please explain how to do this? Thank you.
Find which two sides are perpendicular.
Rotate the triangle about one of those sides.
I tested out the theories A and B. None of them worked. So what does is mean by the x-axis and the y-axis? What does that mean?
Should show your work. Let the points (0,0), (2,1),(1,3) be A,B,C
AB has slope 1/2
AC has slope 3
BC has slope -2
So, AB and BC are perpendicular.
So, rotate the triangle about AB or BC
AB is the line y = 1/2 x or x=2y
BC is the line y = -2x+5
So, A is the correct answer.
Check that against your calculations. You might also try actually plotting the points!
Oh ok! Thank you
To determine about which line the triangle could be resolved to form one cone, you need to examine the coordinates of the given triangle's vertices and the equations of the lines.
First, let's write the equation of each line:
A) the line at x = 2y
B) the line at y = 3x
C) the x-axis
D) the y-axis
Now, let's analyze the points of the triangle and see which line they are closest to:
(0,0): This point lies on both the x and y-axes, so it is equidistant from all the lines.
(2,1): For this point, let's substitute the coordinates into the equations of each line and calculate the distances:
A) Substitute (2,1) into x = 2y:
2 = 2(1)
2 = 2 --> This equation is true, so the point lies on the line x = 2y.
B) Substitute (2,1) into y = 3x:
1 = 3(2)
1 = 6 --> This equation is false, so the point does not lie on the line y = 3x.
C) Substitute (2,1) into the x-axis equation (y = 0):
1 = 0 --> This equation is false, so the point does not lie on the x-axis.
D) Substitute (2,1) into the y-axis equation (x = 0):
2 = 0 --> This equation is false, so the point does not lie on the y-axis.
(1,3): Repeat the above calculations for this point, substituting (1,3) into each equation:
A) Substitute (1,3) into x = 2y:
1 = 2(3)
1 = 6 --> This equation is false, so the point does not lie on the line x = 2y.
B) Substitute (1,3) into y = 3x:
3 = 3(1)
3 = 3 --> This equation is true, so the point lies on the line y = 3x.
C) Substitute (1,3) into the x-axis equation (y = 0):
3 = 0 --> This equation is false, so the point does not lie on the x-axis.
D) Substitute (1,3) into the y-axis equation (x = 0):
1 = 0 --> This equation is false, so the point does not lie on the y-axis.
Based on the above analysis, the triangle can be resolved to form one cone about line B, which is the line y = 3x.