Question

if c is rotated -270° about the point (-1, -2), then reflected along the x - axis, what are the coordinates of c? if e is reflected in the line y = -4.5, then rotated -90° about the point (-1,1), what are the coordinates of e? if a is reflected in the line y = x, then rotated 270° ccw about the point (-1, 2), what are the coordinates of a?

Explanation:

Step1: Analyze rotation -270° about a point

A -270° rotation is equivalent to a 90° clock - wise rotation. The rule for rotating a point $(x,y)$ 90° clock - wise about a center $(a,b)$ is $(x',y')=(a+(y - b),b-(x - a))$. Let the coordinates of point $C$ be $(x,y)$. After rotating point $C$ 90° clock - wise about the point $(-1,-2)$, we get new coordinates $(x_1,y_1)$ using the formula: $x_1=-1+(y + 2)$ and $y_1=-2-(x + 1)$.

Step2: Analyze reflection along the x - axis

The rule for reflecting a point $(x,y)$ along the x - axis is $(x,y)\to(x,-y)$. So, if the point after rotation is $(x_1,y_1)$, the point after reflection $C'$ has coordinates $(x_1,-y_1)$.

However, since the initial coordinates of point $C$ are not given in the problem statement, we cannot calculate the exact numerical answer.

Let's assume the initial coordinates of point $C$ are $(x_0,y_0)$.

1. Rotation step:

• For a 90° clock - wise rotation about the point $(-1,-2)$:
• $x_1=-1+(y_0 + 2)=y_0 + 1$
• $y_1=-2-(x_0 + 1)=-x_0-3$

1. Reflection step:

• After reflection along the x - axis, the coordinates of $C'$ are $(y_0 + 1,x_0 + 3)$.

We need the initial coordinates of $C$ to get a specific numerical answer. For the general case of transformation rules:

• Rotation: If we have a point $(x,y)$ and we rotate it 90° clock - wise about $(a,b)$, we first translate the point so that the center of rotation is at the origin. The translated point is $(x - a,y - b)$. After a 90° clock - wise rotation, it becomes $(y - b,-(x - a))$. Then we translate back, getting $(a+(y - b),b-(x - a))$.
• Reflection: Reflecting $(x,y)$ across the x - axis gives $(x,-y)$.

If we assume $C=(0,0)$ for illustration purposes:

Step1: Rotation

• $x_1=-1+(0 + 2)=1$
• $y_1=-2-(0 + 1)=-3$

Step2: Reflection

• The coordinates of $C'$ are $(1,3)$

Answer:

The coordinates of $C'$ depend on the initial coordinates of $C$. If $C=(x,y)$, after rotation 90° clock - wise about $(-1,-2)$ and reflection along the x - axis, $C'=(y + 1,x + 3)$. If $C=(0,0)$, $C'=(1,3)$