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 phútTEKS 2A,2B,3A,3B,3C,3D幾何学
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
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.
Translate a point
Point P(4, -2) is translated 3 units left and 5 units up. What are the coordinates of P'?
Translation: (x - 3, y + 5) = (4 - 3, -2 + 5) = (1, 3). Move left subtracts from x; move up adds to y.
Reflect a triangle
A triangle has vertices A(2, 3), B(6, 3), and C(4, 7). The triangle is reflected across the y-axis. What are the coordinates of A' (the image of A)?
Reflection across the y-axis changes (x, y) to (-x, y). So A(2, 3) becomes A'(-2, 3). The y-coordinate stays the same; only the x-coordinate sign changes.
Rotate 90° clockwise
A figure with vertices A(1, 2), B(4, 2), C(4, 5) is rotated 90° clockwise about the origin. What are the coordinates of B'?
Rotation 90° clockwise about the origin: (x, y) → (y, -x). So B(4, 2) → B'(2, -4).
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.