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 take a point on the triangle, reflect it over \( y = -x \) (the rule is \( (a,b) \to (-b,-a) \)), and see if the reflected point is also on the triangle. Let's identify the vertices of the right triangle. From the graph, the vertices seem to be \( (0,0) \), \( (4,6) \), and \( (-8,-4) \) (or similar, but visually, reflecting over \( y = -x \) should map the triangle onto itself. Let's test a vertex: take \( (4,6) \), reflect over \( y = -x \): \( (-6,-4) \). Wait, maybe better to see the shape. The line \( y = -x \) passes through the origin, and the triangle appears symmetric with respect to this line (since the blue triangle, when reflected over \( y=-x \), would overlap). So reflective symmetry over \( y = -x \) is Yes.

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

Reflective symmetry over \( x \)-axis means reflecting a point \( (a,b) \) to \( (a,-b) \). Take a point on the triangle, say \( (4,6) \). Reflecting over \( x \)-axis gives \( (4,-6) \), which is not on the blue triangle (the triangle is above and below? Wait, the blue triangle has parts in positive \( y \) and negative \( y \)? Wait, looking at the graph, the blue triangle: one vertex at origin, one at \( (4,6) \), one at \( (-8,-4) \)? Wait, no, maybe the vertices are \( (0,0) \), \( (4,6) \), and \( (-8,-4) \)? Wait, no, the grid: let's check coordinates. The right triangle: from origin, going to \( (4,6) \), and to \( (-8,-4) \)? Wait, maybe better to see that reflecting over \( x \)-axis would flip the \( y \)-coordinate. The triangle has points above and below \( x \)-axis? Wait, the blue area: above \( x \)-axis (from \( y=0 \) to \( y=6 \)) and below \( x \)-axis (from \( y=0 \) to \( y=-4 \))? Wait, no, maybe the vertices are \( (0,0) \), \( (4,6) \), and \( (-8,-4) \). Reflecting \( (4,6) \) over \( x \)-axis is \( (4,-6) \), not on the triangle. Reflecting \( (-8,-4) \) over \( x \)-axis is \( (-8,4) \), not on the triangle. So the triangle is not symmetric over \( x \)-axis. So reflective symmetry over \( x \)-axis is No.

Answer:

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