Transformations and congruence (Geometry) · lesson

Translations slide every point

💡 A translation adds the same amount to every x and the same amount to every y.

A translation slides a figure without turning it. Left and right change x. Up and down change y.

Moving right adds to x, and moving left subtracts. Moving up adds to y, and moving down subtracts.

A rule like (x, y) → (x + 4, y − 1) says the same thing in symbols: 4 right and 1 down.

🧠 Let's work it out

Move (−1, 4) 6 units right and 3 units down.

Right 6 changes x, so −1 + 6 = 5.
Down 3 changes y, so 4 − 3 = 1.
The image is (5, 1).

What do you think comes next?

translation: a slide: every point moves the same distance in the same direction
image: where a point or figure lands after a move

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Reflections flip across a line

💡 A reflection keeps the coordinate along the mirror line and flips the other one.

Across the x-axis, x stays and y changes sign: (x, y) → (x, −y).

Across the y-axis, y stays and x changes sign: (x, y) → (−x, y).

Across the line y = x, the two numbers swap places: (x, y) → (y, x). When a problem lists two moves, do them in the order given.

🧠 Let's work it out

Reflect (−3, 8) across y = x, then move it 4 units up.

Reflecting across y = x swaps the numbers, so (−3, 8) → (8, −3).
Up 4 adds to y, so −3 + 4 = 1.
The final point is (8, 1).

What do you think comes next?

reflection: a flip across a line, like a mirror image
x-axis: the flat number line across the middle of the grid
y-axis: the up-and-down number line through the middle of the grid

Rotations turn about a point

💡 To rotate about a point that is not the origin, move the center to the origin, turn, then move back.

About the origin, a 90° counterclockwise turn sends (x, y) to (−y, x). A 180° turn sends (x, y) to (−x, −y). A 270° counterclockwise turn sends (x, y) to (y, −x).

Counterclockwise means the opposite way a clock's hands move. Unless a problem says clockwise, turns are counterclockwise.

If the center is some other point, first subtract the center from the point. Turn that new point with the rule. Then add the center back.

🧠 Let's work it out

Rotate (6, 2) by 90° counterclockwise about (1, 1).

Subtract the center: (6 − 1, 2 − 1) = (5, 1).
Turn with (x, y) → (−y, x): (5, 1) → (−1, 5).
Add the center back: (−1 + 1, 5 + 1) = (0, 6).

What do you think comes next?

rotation: a turn of a figure about a fixed point
counterclockwise: turning the opposite way from a clock's hands
origin: the point (0, 0)

Congruent figures have matching parts

💡 Slides, flips, and turns keep every length and angle, so matching parts of congruent figures are equal.

Translations, reflections, and rotations are rigid motions. They move a figure but never stretch it. The image is congruent to the original: same size, same shape.

A dilation is different. It grows or shrinks the figure, so lengths change. A dilation with any scale factor other than 1 makes an image that is not congruent.

Congruence statements match letters in order. In triangle ABC ≅ triangle DEF, A matches D, B matches E, and C matches F. Side BC matches side EF.

🧠 Let's work it out

Triangle JKL ≅ triangle MNP, with letters matched in order. Angle J = 45° and angle K = 100°. Find angle P.

P matches L.
The angles of JKL add to 180°, so L = 180 − 45 − 100 = 35°.
Matching angles are equal, so P = 35°.

What do you think comes next?

congruent: exactly the same size and shape
rigid motion: a move that keeps all lengths and angles: a slide, flip, or turn
dilation: a move that makes a figure bigger or smaller

Shortcuts for congruent triangles

💡 SSS, SAS, ASA, and AAS each lock in one triangle, but SSA does not.

You don't need all six parts to know two triangles are congruent. Three side lengths (SSS) build only one triangle. Two sides and the angle between them (SAS) also build only one.

Two angles and the side between them (ASA) work too. So do two angles and a side that is not between them (AAS).

Two sides and an angle that is not between them (SSA) can build two different triangles. So SSA is not enough. Three angles (AAA) only fix the shape, not the size.

🧠 Let's work it out

Ask: could these facts build two different triangles?

Picture the given angle with one side fixed.
Swing the second side. If it can touch the base in two places, two triangles fit.
Two triangles means the facts don't prove congruence. That is what happens with SSA.

What do you think comes next?

included angle: the angle between two given sides
SAS: side, angle, side: two sides and the angle between them