Question

pre-algebra ic sem 2 fall 2025
semester a
02:47:37
which statement correctly describes the diagram?
triangle b is a
reflection of triangle a
across the y - axis.
triangle c is not a
reflection of triangle a.
triangle b is a
reflection of triangle a
across the x - axis.
triangle c is a
reflection of triangle a
across the line
$y = x + 3$.
triangle b is a
reflection of triangle a
across the y - axis.
triangle c is a
reflection of triangle a
across the line
$y = x + 3$
triangle b is a
reflection of triangle a
across the x - axis.
triangle c is not a
reflection of triangle a.

Explanation:

Step1: Analyze reflection across \(x -\)axis

When a point \((x,y)\) is reflected across the \(x -\)axis, the transformation rule is \((x,y)\to(x, - y)\).
For triangle \(A\), if we assume a vertex (say the right - angled vertex) at \((4,2)\), after reflection across the \(x -\)axis, it should be at \((4,-2)\). Looking at triangle \(B\), its right - angled vertex is at \((4, - 2)\). So triangle \(B\) is a reflection of triangle \(A\) across the \(x -\)axis.

Step2: Analyze reflection across \(y=x + 3\)

The general rule for reflection across the line \(y=x + c\) is more complex. But we can use a simple check. Let's take a point \((x_1,y_1)\) in triangle \(A\). Suppose a non - right - angled vertex (say \((2,2)\) in triangle \(A\)).
The formula for reflection across the line \(y=x + 3\):
First, find the perpendicular distance from the point \((x_1,y_1)\) to the line \(y=x+3\) (or \(x - y+3 = 0\)). The distance \(d=\frac{\vert x_1 - y_1+3\vert}{\sqrt{1^2+( - 1)^2}}=\frac{\vert2 - 2 + 3\vert}{\sqrt{2}}=\frac{3}{\sqrt{2}}\).
The direction vector of the line \(y=x + 3\) (or \(x - y+3 = 0\)) is \(\vec{v}=(1,1)\).
The reflection formula:
Let \(a = 1,b=-1,c = 3\). The formula for reflection of a point \((x,y)\) across the line \(ax+by + c=0\) is \(x'=x-\frac{2a(ax + by + c)}{a^{2}+b^{2}}\), \(y'=y-\frac{2b(ax + by + c)}{a^{2}+b^{2}}\).
For \(x = 2,y = 2,a = 1,b=-1,c = 3\):
\(x'=2-\frac{2\times1\times(2-2 + 3)}{1 + 1}=2 - 3=-1\), \(y'=2-\frac{2\times(-1)\times(2 - 2+3)}{1 + 1}=2+3 = 5\).
If we check the vertices of triangle \(C\) (by counting grid units), we can see that triangle \(C\) is a reflection of triangle \(A\) across the line \(y=x + 3\).
For reflection across \(y -\)axis: The rule is \((x,y)\to(-x,y)\). If we take a vertex of triangle \(A\) (e.g., \((4,2)\)), its reflection across \(y -\)axis should be \((-4,2)\), but triangle \(B\) has vertices not consistent with this rule.

Answer:

Triangle \(B\) is a reflection of triangle \(A\) across the \(x -\)axis. Triangle \(C\) is a reflection of triangle \(A\) across the line \(y=x + 3\).