Back to Exercise: Define transformations formally

Exercises: Define Transformations Formally

Work through each section in order. For problems asking you to state or explain definitions, use precise geometric vocabulary — no coordinates unless specifically requested.

Grade 9·20 problems·~30 min·Common Core Math - HS Geometry·standard·hsg-co-a-4

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Warm-Up: Review What You Know

These problems review geometric vocabulary from earlier lessons.

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

Two lines that go in the same direction.

Two lines in the same plane that never intersect.

Two lines that are the same distance apart at every point.

Two lines that form four right angles when they cross.

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

$\ell$ passes through $P$ , and $\ell$ is parallel to $PP'$ .

$\ell$ is perpendicular to $PP'$ , and $\ell$ passes through the midpoint of $PP'$ .

$\ell$ bisects $PP'$ at any angle, and $\ell$ passes through one endpoint.

$\ell$ is perpendicular to $PP'$ , and $\ell$ passes through $P$ .

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?

All points whose $x$ -coordinate equals $r$ .

All points that form a right angle at O.

All points at distance exactly $r$ from $O$ .

All points that can be connected to O by a straight line.

Fluency Practice

Apply the formal geometric definitions of translation, reflection, and rotation.

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?

$PP'$ is parallel to $AB$ ; the length of $PP'$ equals the length of $AB$ ; $PP'$ points in the same direction as $AB$ .

$PP'$ is perpendicular to $AB$ ; the length of $PP'$ equals the length of $AB$ ; $P$ is the midpoint of $AB$ .

The coordinate rule $(x, y) \to (x + a, y + b)$ is applied to $P$ ; the result equals $P'$ .

$PP'$ bisects $AB$ ; the direction from $P$ to $P'$ is counterclockwise; $PP'$ has any length.

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."

$A line of reflection with point P on the line; P maps to itself.$

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?

P maps to a point on the opposite side of the line at distance 1.

$P$ maps to itself ( $P' = P$ ).

$P$ maps to the foot of the perpendicular from $P$ to $\ell$ .

$P$ maps to the reflection of $\ell$ itself.

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?

$OP = OP'$ , and the straight-line distance from $P$ to $P'$ equals $\theta$ .

$P'$ lies on the perpendicular bisector of $OP$ , and angle $POP' = \theta$ .

$OP = OP'$ (so $P'$ lies on the circle centered at $O$ through $P$ ), and the measure of angle $POP' = \theta$ .

$P'$ lies on the same horizontal line as $P$ , and angle $POP' = \theta$ .

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