Question

Question was provided via image upload.

Explanation:

Step1: Identify Transformation Type

The problem involves translating figures (moving vertices along a vector) and rotating figures (about the origin with a given angle). This falls under Geometry (subfield of Mathematics), focusing on transformations.

Step2: Translating a Figure (e.g., Problem 13)

For a translation vector \((a, b)\), each vertex \((x, y)\) becomes \((x + a, y + b)\).

• Given vertices (e.g., \(D, E, F\)) and vector \((x + 2, y - 2)\), apply \(x \to x + 2\), \(y \to y - 2\) to each coordinate.

Step3: Rotating a Figure (e.g., Problem 15)

For a \(180^\circ\) rotation about the origin, the rule is \((x, y) \to (-x, -y)\).

• Take each vertex \((x, y)\), compute \(-x\) and \(-y\) to find new coordinates.

Step4: Combining Transformations (Problem 17)

First rotate \(180^\circ\) (rule \((x, y) \to (-x, -y)\)), then reflect over the \(x\)-axis (rule \((x, y) \to (x, -y)\)).

• Apply rotation first: \((x, y) \to (-x, -y)\), then reflect: \((-x, -y) \to (-x, y)\).

Answer:

To solve these transformation problems:

Translation (e.g., Problem 13):

For vertices \(D, E, F\) and vector \((x + 2, y - 2)\):

• \(D'\): Apply \(x + 2\), \(y - 2\) to \(D\)’s coordinates.
• \(E'\): Same for \(E\).
• \(F'\): Same for \(F\).

Rotation (\(180^\circ\), e.g., Problem 15):

For vertices \(Q, R, S\):

• \(Q'\): \((x, y) \to (-x, -y)\).
• \(R'\): Same rule.
• \(S'\): Same rule.

Combined Transformation (Problem 17):

1. Rotate \(180^\circ\): \((x, y) \to (-x, -y)\) for \(A, B, C\).
2. Reflect over \(x\)-axis: \((-x, -y) \to (-x, y)\) for rotated points.

(Note: Specific coordinates depend on the original figure’s vertices, which are partially visible. Use the transformation rules above with given vertex coordinates to find new points.)