Question

the following diagram shows a right triangle.
which of the following reflective symmetries apply to the triangle?
symmetry\tapplies to the figure?
reflective symmetry over the line $y = -x$\tyes/no
reflective symmetry over the $x$-axis\tyes/no

Explanation:

Step1: Analyze symmetry over \( y = -x \)

To check reflective symmetry over \( y = -x \), we can observe the coordinates of the triangle's vertices. Let's identify the vertices (by looking at the grid): the triangle has vertices, for example, at (0,0), (4,6), and (-8,-4) (approximate, but the key is the reflection over \( y = -x \) swaps \( x \) and \( y \) and negates them, i.e., \((a,b) \to (-b,-a)\)). By visually inspecting the graph, the triangle appears to map onto itself when reflected over \( y = -x \) (since the line \( y = -x \) passes through the triangle in a way that the two sides are mirror images over this line). So the answer for this is Yes.

Step2: Analyze symmetry over \( x \)-axis

Reflective symmetry over the \( x \)-axis means that for every point \((x,y)\) on the figure, \((x,-y)\) is also on the figure. Looking at the triangle, the upper part (positive \( y \)) and lower part (negative \( y \)) do not mirror each other. For example, the vertex at (4,6) would need a corresponding vertex at (4,-6) to have \( x \)-axis symmetry, but there's no such vertex. So the answer for this is No.

Answer:

Reflective symmetry over the line \( y = -x \): Yes
Reflective symmetry over the \( x \)-axis: No