Question

plot the image of triangle $\triangle abc$ under a reflection across line $\ell$.

Explanation:

Step1: Find reflection of A

Count the vertical distance from A to line \( \ell \). Let's say A is \( d \) units above \( \ell \). Plot \( A' \) \( d \) units below \( \ell \) (since reflection over horizontal line, vertical distance is preserved, opposite side).

Step2: Find reflection of B

Similarly, find vertical distance from B to \( \ell \), plot \( B' \) on the other side of \( \ell \) at same distance.

Step3: Find reflection of C

Find vertical distance from C to \( \ell \), plot \( C' \) on the other side of \( \ell \) at same distance.

Step4: Connect \( A' \), \( B' \), \( C' \)

Connect the reflected points to form \( \triangle A'B'C' \), the image of \( \triangle ABC \) under reflection across \( \ell \).

(Note: Since it's a grid, we can use coordinates. Let's assume line \( \ell \) is horizontal (x - axis - like). For example, if A is at (x1, y1), reflection over \( \ell \) (y = k) is (x1, 2k - y1). We calculate for each point:

• Let's find coordinates (assuming grid lines are 1 unit). Suppose \( \ell \) is the horizontal line (say y = 0 for the bottom - most horizontal line shown).
• Point A: Let's say A is at (x_A, y_A). If A is 3 units above \( \ell \), then \( A' \) is 3 units below \( \ell \) (same x - coordinate, y - coordinate = \( y_{\ell}-(y_A - y_{\ell}) \)).
• Do the same for B and C, then connect.)

Answer:

To plot the reflection:

1. For each vertex \( A \), \( B \), \( C \) of \( \triangle ABC \), draw a perpendicular line to line \( \ell \) (which is horizontal here) passing through the vertex.
2. Measure the distance from the vertex to line \( \ell \) along this perpendicular.
3. Mark a point on the other side of line \( \ell \) at the same distance from \( \ell \) as the original vertex. These points are \( A' \), \( B' \), \( C' \) (the reflections of \( A \), \( B \), \( C \) respectively).
4. Connect \( A' \), \( B' \), and \( C' \) to form the reflected triangle \( \triangle A'B'C' \).

(If we use coordinates with \( \ell \) as \( y = k \), the reflection of a point \( (x,y) \) over \( y = k \) is \( (x, 2k - y) \). We apply this formula to each vertex of \( \triangle ABC \) to get the reflected vertices and then plot them.)