Question

plot the image of triangle △abc under a reflection across line ℓ.

Explanation:

Step1: Recall reflection rule

For a point $(x,y)$ reflected across a vertical line $x = a$, the new - x - coordinate is $2a - x$ and the y - coordinate remains the same.

Step2: Reflect point A

If point A is on the line of reflection, its image is itself. Let's assume the line of reflection $\ell$ is a vertical line. If the x - coordinate of A is $x_A$ and the equation of the line $\ell$ is $x = a$ (and since A is on $\ell$, $x_A=a$), the image of A, $A'=A$.

Step3: Reflect point B

Measure the horizontal distance from point B to the line of reflection $\ell$. Let the x - coordinate of B be $x_B$ and the equation of the line $\ell$ be $x = a$. The horizontal distance is $d=x_B - a$. The x - coordinate of the image of B, $B'$ is $a-(x_B - a)=2a - x_B$, and the y - coordinate of $B'$ is the same as that of B.

Step4: Reflect point C

Similarly, for point C with x - coordinate $x_C$, measure the horizontal distance from C to the line of reflection $\ell$. Let the equation of $\ell$ be $x = a$. The x - coordinate of the image of C, $C'$ is $2a - x_C$, and the y - coordinate of $C'$ is the same as that of C.

Step5: Plot the new triangle

Connect the reflected points $A'$, $B'$, and $C'$ to form the reflected triangle $\triangle A'B'C'$.

Since this is a visual - based problem and we can't actually plot in this text - based format, the general steps for reflection are as above. To actually solve this, one would use a ruler and graph paper to accurately measure the distances and plot the points.

(Note: In a real - world scenario, if we assume the line of reflection is the y - axis ($x = 0$), for a point $(x,y)$ its reflection is $(-x,y)$. We would apply this rule to each vertex of $\triangle ABC$ to get the reflected points and then plot them.)

Answer:

Step1: Recall reflection rule

For a point $(x,y)$ reflected across a vertical line $x = a$, the new - x - coordinate is $2a - x$ and the y - coordinate remains the same.

Step2: Reflect point A

If point A is on the line of reflection, its image is itself. Let's assume the line of reflection $\ell$ is a vertical line. If the x - coordinate of A is $x_A$ and the equation of the line $\ell$ is $x = a$ (and since A is on $\ell$, $x_A=a$), the image of A, $A'=A$.

Step3: Reflect point B

Measure the horizontal distance from point B to the line of reflection $\ell$. Let the x - coordinate of B be $x_B$ and the equation of the line $\ell$ be $x = a$. The horizontal distance is $d=x_B - a$. The x - coordinate of the image of B, $B'$ is $a-(x_B - a)=2a - x_B$, and the y - coordinate of $B'$ is the same as that of B.

Step4: Reflect point C

Similarly, for point C with x - coordinate $x_C$, measure the horizontal distance from C to the line of reflection $\ell$. Let the equation of $\ell$ be $x = a$. The x - coordinate of the image of C, $C'$ is $2a - x_C$, and the y - coordinate of $C'$ is the same as that of C.

Step5: Plot the new triangle

Connect the reflected points $A'$, $B'$, and $C'$ to form the reflected triangle $\triangle A'B'C'$.

Since this is a visual - based problem and we can't actually plot in this text - based format, the general steps for reflection are as above. To actually solve this, one would use a ruler and graph paper to accurately measure the distances and plot the points.

(Note: In a real - world scenario, if we assume the line of reflection is the y - axis ($x = 0$), for a point $(x,y)$ its reflection is $(-x,y)$. We would apply this rule to each vertex of $\triangle ABC$ to get the reflected points and then plot them.)