Question

select the sequence of transformations that maps triangle lap onto triangle lap.
a triangle lap was rotated 90 degrees clockwise to triangle lap.
b triangle lap was translated 1 unit to the left and 3 units up to triangle lap.
c triangle lap was translated 1 unit to the right and 3 units down to triangle lap.
d triangle lap was reflected over the x - axis to triangle lap.

Explanation:

Step1: Analyze translation

Translation is a transformation that slides a figure. Let's check the movement of the vertices.
Let's assume the coordinates of \(L\), \(A\), \(P\) and \(L'\), \(A'\), \(P'\). If we consider the general rule of translation \((x,y)\to(x + h,y + k)\).
For example, if we take a vertex (say \(L\)). Suppose \(L\) has coordinates \((x_1,y_1)\) and \(L'\) has coordinates \((x_2,y_2)\). By counting the units on the coordinate - plane, we find that for each vertex:
If we consider the horizontal movement ( \(x\) - direction) and vertical movement ( \(y\) - direction).
The horizontal movement: we can see that the figure moves to the right. If we count the units, the horizontal shift \(h = 1\) (right - direction is positive for \(x\) - axis).
The vertical movement: the figure moves down. If we count the units, the vertical shift \(k=- 3\) (down - direction is negative for \(y\) - axis).

Step2: Check other transformations

• Rotation:

A \(90^{\circ}\) clockwise rotation about the origin has the rule \((x,y)\to(y, - x)\). If we apply this rule to the vertices of \(\triangle LAP\), the resulting figure will not match \(\triangle L'A'P'\) in terms of position.

• Reflection over the \(x\) - axis:

The rule for reflection over the \(x\) - axis is \((x,y)\to(x, - y)\). Applying this rule to the vertices of \(\triangle LAP\) will not give the position of \(\triangle L'A'P'\) as seen in the coordinate - plane.

• Incorrect translation (Option B):

Option B says 1 unit to the left (\(h=-1\)) and 3 units up (\(k = 3\)), which is not what we observe from the movement of vertices from \(\triangle LAP\) to \(\triangle L'A'P'\)

Answer:

C. Triangle LAP was translated 1 unit to the right and 3 units down to triangle L'A'P'.