Question

the coordinates of $\triangle abc$ are $a(1, 2)$, $b(3, 2)$, and $c(2, 5)$. a transformation maps it onto $\triangle abc$ with coordinates $a(-1, 2)$, $b(-3, 2)$, and $c(-2, 5)$. the triangles are congruent by sss. which transformation shows the triangles are congruent? \\(\bigcirc\\) a. a rotation of $90^\circ$ counterclockwise about the origin \\(\bigcirc\\) b. a reflection over the $x$-axis \\(\bigcirc\\) c. a rotation of $180^\circ$ about the origin \\(\bigcirc\\) d. a reflection over the $y$-axis \\(\bigcirc\\) e. a reflection over the line $y = x$

Explanation:

Step1: Recall reflection over y - axis rule

The rule for a reflection over the \(y\) - axis is \((x,y)\to(-x,y)\).

Step2: Check coordinates of A

For point \(A(1,2)\), applying the reflection over \(y\) - axis rule: \((1,2)\to(- 1,2)\), which matches \(A'(-1,2)\).

Step3: Check coordinates of B

For point \(B(3,2)\), applying the reflection over \(y\) - axis rule: \((3,2)\to(-3,2)\), which matches \(B'(-3,2)\).

Step4: Check coordinates of C

For point \(C(2,5)\), applying the reflection over \(y\) - axis rule: \((2,5)\to(-2,5)\), which matches \(C'(-2,5)\).

Step5: Eliminate other options

• Option a: Rotation of \(90^{\circ}\) counter - clockwise about the origin has the rule \((x,y)\to(-y,x)\). For \(A(1,2)\), it would be \((-2,1)

eq(-1,2)\), so this is incorrect.

• Option b: Reflection over \(x\) - axis has the rule \((x,y)\to(x, - y)\). For \(A(1,2)\), it would be \((1,-2)

eq(-1,2)\), so this is incorrect.

• Option c: Rotation of \(180^{\circ}\) about the origin has the rule \((x,y)\to(-x,-y)\). For \(A(1,2)\), it would be \((-1,-2)

eq(-1,2)\), so this is incorrect.

• Option e: Reflection over the line \(y = x\) has the rule \((x,y)\to(y,x)\). For \(A(1,2)\), it would be \((2,1)

eq(-1,2)\), so this is incorrect.

Answer:

D. a reflection over the y - axis