Exercises: Define Transformations Formally

Which statement gives the precise geometric definition of parallel lines (from HSG.CO.A.1)?

The perpendicular bisector of segment $PP'$ is a line $\ell$ that satisfies two conditions. Which pair correctly states both?

According to the CO.A.1 definition, a circle centered at point $O$ with radius $r$ is the set of all points in the plane that satisfy which condition?

The formal definition of a translation along directed segment $\overrightarrow{AB}$ states that each point $P$ maps to $P'$ such that segment $PP'$ satisfies which three conditions?

Complete the formal definition of a translation: "The translation along $\overrightarrow{AB}$ maps each point $P$ to the point $P'$ such that $PP'$ is   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   to $AB$, the length of $PP'$   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   the length of $AB$, and $PP'$ and $AB$ point in the   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   direction."

The formal definition of a reflection across line $\ell$ states that $P$ maps to $P'$ such that $\ell$ is the perpendicular bisector of $PP'$. For a point $P$ that lies directly on line $\ell$, where does $P$ map?

For a reflection across line $\ell$: if point $P$ is not on $\ell$, then $P$ maps to $P'$ such that $\ell$ is   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   to $PP'$ and passes through the   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   of $PP'$.

The formal definition of a rotation by angle $\theta$ about center $O$ states that $P$ maps to $P'$ such that two conditions hold (when $P \neq O$). Which pair correctly states both conditions?

Which CO.A.1 geometric term is the central building block of the formal definition of a translation?

Point $P = (4, 1)$ is reflected across line $\ell: x = 0$ (the $y$-axis). Using the formal definition of reflection, which statement correctly identifies where $P'$ is and why?

Center $O$ is at the origin. Point $P = (3, 0)$ is rotated $90^\circ$ counterclockwise about $O$ to land at $P' = (0, 3)$. A classmate says: "Point $P$ moved in a straight line from $(3, 0)$ to $(0, 3)$." Use the formal definition of rotation to explain why this is incorrect. Your answer must reference the circle centered at $O$.

Triangle $ABC$ is rotated $60^\circ$ about point $O$, which is located 10 units to the left of the triangle (entirely outside the triangle). Which statement about this rotation is correct according to the formal definition?

Which set of CO.A.1 terms appears in the formal definition of a translation (and not in the definitions of reflection or rotation)?

Explain why the student's claim is incorrect. Your answer must describe one situation in which the coordinate rule fails to define the translation, and state what a geometric definition provides that the coordinate rule does not.

Explain what is correct and what is imprecise about Sofia's description, using the formal definition of reflection. Your answer must identify the CO.A.1 term that is the geometric heart of the reflection definition.

Using the formal definition of reflection, is Priya's verification complete?

What is the fundamental error in Alex's definition?

What is wrong with Jordan's definition of reflection?

Center $O$ is at the origin. Point $P = (5, 0)$ is rotated $90^\circ$ counterclockwise about $O$.

Part (a): Use the formal definition of rotation to find the coordinates of $P'$. Show the two conditions — equal radii and the angle condition — in your work.

Part (b): A classmate says: "Since $OP = 5$ and the rotation is $90^\circ$, the chord length $PP'$ must also equal 5." Is the classmate correct? Compute the chord length $|PP'|$ and explain what determines it.

The summary table from the lesson states: "Together, the three rigid motion definitions use all five CO.A.1 geometric terms: angle, circle, perpendicular line, parallel line, and line segment."

Verify this claim by listing each of the five CO.A.1 terms and identifying precisely where each term appears in one of the three formal definitions (translation, reflection, or rotation). Then explain in one sentence why this architectural connection between CO.A.1 and CO.A.4 matters for the proof work in CO.B.