Transformations: Sliding, Flipping & Turning Figures

Geometric transformations are just ways of moving a figure around the coordinate plane without redrawing it. There are three basic moves: slide it to a new spot (translation), flip it over a line (reflection), or spin it around a point (rotation). Each one changes a point's coordinates according to a fixed rule — learn the rules and you can transform any figure by transforming its vertices.

This guide gives you the coordinate rule for each transformation, with worked and practice problems matched to real test difficulty at Northside Tutoring.

Why Transformations Matter

Transformations appear on the SAT and ACT as coordinate-rule questions and as the conceptual backbone of congruence and symmetry. They reward knowing a handful of rules cold. They're beyond the SSAT but connect to the function transformations you meet in vertex form.

Translations: Sliding

A translation slides every point the same distance in the same direction — no flipping or turning. Add to the coordinates:

"Right 3 and down 2" means (x + 3, y − 2). Apply it to each vertex to move the whole figure.

Reflections: Flipping

A reflection flips a figure over a line (the mirror). The common rules:

Reflect across…	Rule
the x-axis	(x, y) → (x, −y)
the y-axis	(x, y) → (−x, y)
the line y = x	(x, y) → (y, x)

Seeing a Reflection

Rotations: Turning

A rotation turns a figure about a point — usually the origin. The counterclockwise rules:

Rotate (about origin)	Rule
90° counterclockwise	(x, y) → (−y, x)
180°	(x, y) → (−x, −y)
270° counterclockwise	(x, y) → (y, −x)

Rigid Motions Preserve Size

Translations, reflections, and rotations are rigid motions: they move a figure without changing its size or shape, so the image is always congruent to the original. (Dilations, which scale a figure, are the exception — they change size and produce a similar figure.)

Where You'll See This — Test by Test

No reference sheet covers the rules — memorize them. The SAT and ACT test coordinate transformations and which motions preserve congruence. They're beyond the SSAT.

Watch the Lesson

Sometimes a diagram needs a voice. In the short video below, one of our Northside tutors walks through the core idea and works through test-style problems in real time.

Worked Example Problems

These problems are calibrated to the difficulty you'll actually see on test day. Try each one before opening the solution.

Common Mistakes to Avoid

Practice Problems — You Try

Three problems below. Work each before checking the solution.

The Northside Method — How We Teach This 1-on-1

Reading a blog is a great starting point. But there's a meaningful gap between understanding a concept and reflexively applying it under timed conditions. That gap is exactly what our tutors close.

Every Northside student works through a four-step framework:

1. Assessment. We diagnose which specific skills are slowing your student down — not just whether they "get it" in the abstract.
2. Perfect-match coach. We pair them with an elite tutor (we accept only the top 1% of applicants) whose teaching style fits how your student actually learns.
3. Bespoke plan. A roadmap built around your student's target score, target timeline, and current pacing data.
4. Data-driven adjustment. Every session ends with a check on whether the student's accuracy and speed are moving in the right direction.

And if a student meets all eligibility requirements but doesn't hit the defined score improvement? We provide 5 additional hours of cohort learning at no cost. That's the Northside guarantee — built on 25 years of measured outcomes.