Answered

Unit Post Test

When graphed, the three lines [tex]y = -x + 2[/tex], [tex]y = 2x - 1[/tex], and [tex]y = x - 2[/tex] intersect in such a way that they form a triangle. What are the coordinates of the three vertices of this triangle?

A. [tex](2, 0), (0, 2)[/tex], and [tex](-1, -3)[/tex]
B. [tex](0, 2), (2, 0)[/tex], and [tex](1, -1)[/tex]
C. [tex](1, 1), (2, 0)[/tex], and [tex](-1, -3)[/tex]
D. [tex](1, 1), (0, 2)[/tex], and [tex](-1, -3)[/tex]
E. [tex](2, 0), (1, -1)[/tex], and [tex](-1, -3)[/tex]



Answer :

GinnyAnswer
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]