학습 · 기하학

Coordinate Geometry & Transformations: Move, Flip, Spin, Scale

Translate, reflect, rotate, dilate — the four moves on the coordinate plane and the rules for each. Plus the distance, midpoint, and slope formulas you need on every coordinate question.

9 분 TEKS 2A,2B,3A,3B,3C,3D 기하학

The grid is your friend

On the coordinate plane, every “find the missing point” or “what shape is this” question becomes plug-and-chug. You only need three formulas and four transformation rules — and the CBE tests them in the same way every year.

Three formulas to memorize

Distance: d = √[(x2 − x1)2 + (y2 − y1)2] Midpoint: M = ( (x1+x2)/2 , (y1+y2)/2 ) Slope: m = (y2 − y1) / (x2 − x1) Distance is Pythagorean in disguise. Midpoint is the average. Slope is rise over run.
Parallel & perpendicular by slope

Parallel lines have equal slopes. Perpendicular lines have negative reciprocal slopes (m and −1/m). Horizontal → m = 0. Vertical → m undefined.

The four transformations

TRANSLATION slide: (x, y) → (x+a, y+b) REFLECTION flip across line: x → −x or y → −y ROTATION spin around point (90°, 180°, 270°) DILATION scale from center: (x, y) → (kx, ky) SUMMARY Translation: same size, same shape Reflection: same size, mirror Rotation: same size, turned Dilation: scaled (size changes) First three preserve congruence; dilation produces similar figures.
Three rigid motions (translation, reflection, rotation) preserve size. Dilation scales it.

The (x, y) rules

Translate by (a, b): (x, y) → (x + a, y + b) Reflect across x-axis: (x, y) → (x, −y) Reflect across y-axis: (x, y) → (−x, y) Reflect across y = x: (x, y) → (y, x) Rotate 90° CCW about origin: (x, y) → (−y, x) Rotate 90° CW about origin: (x, y) → (y, −x) Rotate 180° about origin: (x, y) → (−x, −y) Dilate by factor k (center at origin): (x, y) → (kx, ky)
Rotation memory hook

For 90° CCW (counter-clockwise): swap, then negate the new x. CW: swap, then negate the new y. 180°: negate both. Always start by swapping x and y — the signs come last.

Practice

Translate a point

Point P(4, −2) is translated 3 units left and 5 units up. What are the coordinates of P′?

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Practice

Reflect a triangle

Triangle with vertices A(2, 3), B(6, 3), C(4, 7) reflected across the x-axis. Find the image vertices.

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Practice

Rotate 90° clockwise

A figure with vertices A(1, 2), B(4, 2), C(4, 5) is rotated 90° clockwise about the origin. Find the image.

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Composite transformations

Order matters

If a problem says “reflect, then translate” — do them in that order. Reflecting a translated figure is not the same as translating a reflected figure. Apply each step to the result of the previous step.

3-second recap

  • Distance = √[(Δx)² + (Δy)²]
  • Midpoint = average of x's and y's
  • Slope = Δy / Δx
  • Translation, reflection, rotation → congruent image
  • Dilation → similar image (scale factor k)