Question

triangle abc has coordinates a(2, 0), b(-1, 5), and c(4, 3). determine the coordinates of the vertices of the image after a rotation 180° about the origin.
a) a(2, 2), b(-1, 7), and c(4, 5)
b) a(-2, 0), b(1, -5), and c(-4, -3)
c) a(5, -4), b(2, 1), and c(7, -1)
d) a(0, 2), b(-5, 1), and c(-3, 4)

Explanation:

Step1: Recall 180° rotation rule

For a point \((x, y)\), a 180° rotation about the origin gives \((-x, -y)\).

Step2: Apply to point A(2, 0)

\(A' = (-2, -0) = (-2, 0)\)? Wait, no, wait the original A is (2, 0)? Wait, no, the problem says A(2, 0)? Wait, no, looking at the options, let's re - check. Wait, the original triangle ABC has A(2, 0), B(-1, 5), C(4, 3). Wait, no, the user's image: "Triangle ABC has coordinates A(2, 0), B(-1, 5), and C(4, 3). Determine the coordinates of the image after a rotation 180° about the origin."

Wait, 180° rotation formula: \((x,y)\to(-x,-y)\)

Step3: Rotate A(2, 0)

\(A' = (-2, -0)=(-2, 0)\)? Wait, no, wait the options: Let's check each option. Wait, maybe I misread A's coordinate. Wait, the original A is (2, 0)? Wait, option B is A'(-2, 0), B'(1, -5), C'(-4, -3). Let's rotate each point:

• For \(A(2, 0)\): \(x = 2,y = 0\), so \(A'=(-2,-0)=(-2,0)\)
• For \(B(-1,5)\): \(x=-1,y = 5\), so \(B'=(1,-5)\)
• For \(C(4,3)\): \(x = 4,y = 3\), so \(C'=(-4,-3)\)

Which matches option B.

Answer:

B. \(A'(-2, 0)\), \(B'(1, -5)\), and \(C'(-4, -3)\)