Topic 8 of 8

Transformations, congruence, and similarity

Step 10 · Pre-algebra topics

Translate, reflect, rotate and dilate shapes on the coordinate plane, and use these moves to decide whether two figures are congruent or similar.

Translations, reflections and rotations

A transformation moves a shape. Translations slide it, reflections flip it over a line, and rotations turn it about a point. All three keep every length and every angle the same.

A triangle (dark) and its reflection over the y-axis (light): the corner (4, 1) goes to (−4, 1).

Move	Rule for (x, y)	Example: (3, 2)
translate 4 right and 1 down	(x + 4, y − 1)	(7, 1)
reflect over the x-axis	(x, −y)	(3, −2)
reflect over the y-axis	(−x, y)	(−3, 2)
rotate 90° counterclockwise about (0, 0)	(−y, x)	(−2, 3)
rotate 180° about (0, 0)	(−x, −y)	(−3, −2)

Example: Translating

Point A(1, 2) is translated 3 left and 5 up. Where does it go?

x: 1 − 3 = −2.
y: 2 + 5 = 7.

Answer(−2, 7)

Dilations

A dilation enlarges or shrinks a shape from a center by a scale factor k. With the center at the origin, (x, y) goes to (kx, ky). Angles stay the same, and every length is multiplied by k.

A triangle dilated from the origin by scale factor 2: every corner's coordinates double.

Example: A dilation

Dilate the point (4, −6) from the origin by scale factor $\frac{1}{2}$ .

Multiply both coordinates by $\frac{1}{2}$ .

Answer(2, −3)

Congruent and similar figures

Two figures are congruent if translations, reflections and rotations can take one exactly onto the other: same shape and same size.

Two figures are similar if a dilation is allowed too: same shape, any size. Matching angles are equal, and matching sides are in the same ratio.

Example: A missing side

Triangle ABC is similar to triangle DEF. AB = 4, BC = 6 and DE = 10. How long is EF?

The scale factor from ABC to DEF is 10 ÷ 4 = 2.5.
EF = 6 × 2.5 = 15.

Answer15

Example: Two matching angles

Two triangles each have angles of 40° and 75°. Are they similar?

The third angles are both 180 − 115 = 65°.
All three angles match.

AnswerYes: two matching angles are enough

Try it yourself

Work each one on paper first, then open the answer to check.

Translate (5, −1) by 2 left and 4 up.
Show answer
(3, 3)
Reflect (−3, 7) over the x-axis.
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(−3, −7)
Rotate (4, 1) by 180° about the origin.
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(−4, −1)
Dilate (6, 9) from the origin by scale factor 3.
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(18, 27)
A triangle with sides 3, 4 and 5 is similar to one whose shortest side is 12. What is its longest side?
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20
Do translations and reflections change a shape's area?
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No: they keep every length, so the area stays the same.