Answer :
To find the coordinates of the vertices of the triangle formed by the three lines [tex]\( y = -x + 2 \)[/tex], [tex]\( y = 2x - 1 \)[/tex], and [tex]\( y = x - 2 \)[/tex], we need to determine the points where each pair of lines intersects. Let's go through this step-by-step:
1. Intersection of [tex]\( y = -x + 2 \)[/tex] and [tex]\( y = 2x - 1 \)[/tex]:
- Set the equations equal to each other:
[tex]\[ -x + 2 = 2x - 1 \][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[ 2 + 1 = 2x + x \implies 3 = 3x \implies x = 1 \][/tex]
- Substitute [tex]\( x = 1 \)[/tex] back into either equation to find [tex]\( y \)[/tex]:
[tex]\[ y = 2(1) - 1 = 2 - 1 = 1 \][/tex]
- Therefore, the intersection point is [tex]\( (1, 1) \)[/tex].
2. Intersection of [tex]\( y = -x + 2 \)[/tex] and [tex]\( y = x - 2 \)[/tex]:
- Set the equations equal to each other:
[tex]\[ -x + 2 = x - 2 \][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[ 2 + 2 = x + x \implies 4 = 2x \implies x = 2 \][/tex]
- Substitute [tex]\( x = 2 \)[/tex] back into either equation to find [tex]\( y \)[/tex]:
[tex]\[ y = 2 - 2 = 0 \][/tex]
- Therefore, the intersection point is [tex]\( (2, 0) \)[/tex].
3. Intersection of [tex]\( y = 2x - 1 \)[/tex] and [tex]\( y = x - 2 \)[/tex]:
- Set the equations equal to each other:
[tex]\[ 2x - 1 = x - 2 \][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[ 2x - x = -2 + 1 \implies x = -1 \][/tex]
- Substitute [tex]\( x = -1 \)[/tex] back into either equation to find [tex]\( y \)[/tex]:
[tex]\[ y = 2(-1) - 1 = -2 - 1 = -3 \][/tex]
- Therefore, the intersection point is [tex]\( (-1, -3) \)[/tex].
The coordinates of the vertices of the triangle are:
- [tex]\((1, 1)\)[/tex]
- [tex]\((2, 0)\)[/tex]
- [tex]\((-1, -3)\)[/tex]
Hence, the correct answer is:
C. [tex]\( (1, 1), (2, 0), (-1, -3) \)[/tex]
1. Intersection of [tex]\( y = -x + 2 \)[/tex] and [tex]\( y = 2x - 1 \)[/tex]:
- Set the equations equal to each other:
[tex]\[ -x + 2 = 2x - 1 \][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[ 2 + 1 = 2x + x \implies 3 = 3x \implies x = 1 \][/tex]
- Substitute [tex]\( x = 1 \)[/tex] back into either equation to find [tex]\( y \)[/tex]:
[tex]\[ y = 2(1) - 1 = 2 - 1 = 1 \][/tex]
- Therefore, the intersection point is [tex]\( (1, 1) \)[/tex].
2. Intersection of [tex]\( y = -x + 2 \)[/tex] and [tex]\( y = x - 2 \)[/tex]:
- Set the equations equal to each other:
[tex]\[ -x + 2 = x - 2 \][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[ 2 + 2 = x + x \implies 4 = 2x \implies x = 2 \][/tex]
- Substitute [tex]\( x = 2 \)[/tex] back into either equation to find [tex]\( y \)[/tex]:
[tex]\[ y = 2 - 2 = 0 \][/tex]
- Therefore, the intersection point is [tex]\( (2, 0) \)[/tex].
3. Intersection of [tex]\( y = 2x - 1 \)[/tex] and [tex]\( y = x - 2 \)[/tex]:
- Set the equations equal to each other:
[tex]\[ 2x - 1 = x - 2 \][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[ 2x - x = -2 + 1 \implies x = -1 \][/tex]
- Substitute [tex]\( x = -1 \)[/tex] back into either equation to find [tex]\( y \)[/tex]:
[tex]\[ y = 2(-1) - 1 = -2 - 1 = -3 \][/tex]
- Therefore, the intersection point is [tex]\( (-1, -3) \)[/tex].
The coordinates of the vertices of the triangle are:
- [tex]\((1, 1)\)[/tex]
- [tex]\((2, 0)\)[/tex]
- [tex]\((-1, -3)\)[/tex]
Hence, the correct answer is:
C. [tex]\( (1, 1), (2, 0), (-1, -3) \)[/tex]