draw the following triangle after a $180^{\\circ}$ rotation about the origin.

Explanation:

Step1: Identify triangle vertices

Original vertices: $(1,-2)$, $(3,-5)$, $(5,-4)$

Step2: Apply 180° rotation rule

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

• $(1,-2) \to (-1,2)$
• $(3,-5) \to (-3,5)$
• $(5,-4) \to (-5,4)$

Step3: Plot new vertices and connect

Plot $(-1,2)$, $(-3,5)$, $(-5,4)$; draw lines between them.

Answer:

The rotated triangle has vertices at $(-1,2)$, $(-3,5)$, and $(-5,4)$, connected to form a triangle on the coordinate plane.