Question

4. A triangle has vertices (-2,-3) (3,5) and (8,-1) .

Answer 1

Given:
Vertex 1 (-2,-3)
Vertex 2 (3,5)
Vertex 3 (8,-1)

Reflection across the x-axis rule (x,y) → (x -y)
Vertex 1 (-2,-3) → (-2,-(-3)) → (-2,3)
Vertex 2 (3,5) → (3,-5)
Vertex 3 (8,-1) → (8,-(-1)) → (8,1)

Rotation 90° clockwise (x,y) → (y,-x)  *I'm assuming the original triangle was rotated and not the reflection.
Vertex 1 (-2,-3) → (-3,-(-2)) → (-3,2)
Vertex 2 (3,5) → (5,-3)
Vertex 3 (8,-1) → (-1,-8)

Answer 2

Answer with explanation:

(a)→Coordinates of Vertices of triangle = (-2,-3), (3,5) and (8, -1).

Plotting these points in two dimensional coordinate plane:

Now when the points are reflected through X axis, the distance of points from line , y=0, on both sides is same .That is distance of Preimage from ,line, y=0 is same as Distance of Image from the line.

≡Rule of reflection of (x,y)→ (x, -y).

⇒Coordinates of Image =(-2,3), (3, -5), (8,1).

In Matrix form

$\left[\begin{array}{ccc}-2&-3\\3&5\\8&-1\end{array}\right] \text {reflected across X axis}=\left[\begin{array}{ccc}-2&3\\3&-5\\8&1\end{array}\right]$

(b) → Coordinates of Vertices of triangle = (-2,-3), (3,5) and (8, -1) is rotated by 90 degree, the vertices of triangle will be , (-3, 2), (5, -3), (-1, -8).

The rule is , after 90 degree rotation . point (a,b)→(b, -a).

Plotting these points in two dimensional coordinate plane:

In Matrix form

$\left[\begin{array}{ccc}-2&-3\\3&5\\8&-1\end{array}\right] \text {reflected across X axis}=\left[\begin{array}{ccc}-3&2\\5&-3\\-1&-8\end{array}\right]$