Question

which sequence of transformations will carry △abc onto △abc?
a translation left 3 units and translation up 1 unit
b translation left 1 unit and translation down 3 units
c 180 clockwise rotation about the origin and a reflection about the y - axis
d 180 clockwise rotation about the origin and a reflection about the y = x line

Explanation:

Step1: Analyze translation

Translation involves moving the figure without rotation or reflection. Check the coordinates of corresponding vertices of $\triangle ABC$ and $\triangle A'B'C'$ to see if translation rules match.

Step2: Analyze rotation and reflection

For rotation, use the rules for rotating a point $(x,y)$ 180 - degree clock - wise about the origin: $(x,y)\to(-x,-y)$. For reflection about the $y$ - axis, $(x,y)\to(-x,y)$; for reflection about the $y = x$ line, $(x,y)\to(y,x)$.

Step3: Check option A

If we consider a translation left 3 units and up 1 unit, if a point $(x,y)$ in $\triangle ABC$ is transformed, the new point will be $(x - 3,y + 1)$. By checking the vertices of the triangles, we can see if this transformation maps $\triangle ABC$ onto $\triangle A'B'C'$.

Step4: Check option B

For a translation left 1 unit and down 3 units, a point $(x,y)$ will be transformed to $(x - 1,y-3)$. Check if this maps the vertices correctly.

Step5: Check option C

For a 180 - degree clock - wise rotation about the origin, a point $(x,y)$ becomes $(-x,-y)$. Then for a reflection about the $y$ - axis, $(-x,-y)$ becomes $(x,-y)$. Check if this sequence maps $\triangle ABC$ onto $\triangle A'B'C'$.

Step6: Check option D

For a 180 - degree clock - wise rotation about the origin, a point $(x,y)$ becomes $(-x,-y)$. Then for a reflection about the $y = x$ line, $(-x,-y)$ becomes $(-y,-x)$. Check if this sequence maps $\triangle ABC$ onto $\triangle A'B'C'$.

Let's assume a vertex of $\triangle ABC$ is $(x,y)$.
For option A:
If we have a vertex $(x,y)$ in $\triangle ABC$, after translation left 3 units and up 1 unit, the new vertex is $(x-3,y + 1)$.
For option B:
If we have a vertex $(x,y)$ in $\triangle ABC$, after translation left 1 unit and down 3 units, the new vertex is $(x - 1,y-3)$.
For option C:
If we have a vertex $(x,y)$ in $\triangle ABC$, after 180 - degree clock - wise rotation about the origin, it becomes $(-x,-y)$. After reflection about the $y$ - axis, it becomes $(x,-y)$.
For option D:
If we have a vertex $(x,y)$ in $\triangle ABC$, after 180 - degree clock - wise rotation about the origin, it becomes $(-x,-y)$. After reflection about the $y = x$ line, it becomes $(-y,-x)$.

By observing the position of the triangles on the grid and applying these transformation rules to the vertices, we find that the correct sequence of transformations is a translation left 3 units and translation up 1 unit.

Answer:

A. translation left 3 units and translation up 1 unit