Math Core

Two puzzle pieces cut from the same mold are identical, while a photo and its wallet-size print have the same shape but different sizes. Transformations give you a precise way to say when two figures are "the same": you ask which moves carry one figure onto the other.

Congruent figures

Definition

Congruent

Two figures are congruent if one can be carried onto the other by a sequence of rigid motions: translations, reflections and rotations. The symbol $\cong$ means "is congruent to."

Rigid motions never change lengths or angles, so congruent figures have the same size and the same shape. Their corresponding parts (the matching sides and angles) are equal.

The order of the letters matters. $\triangle ABC \cong \triangle DEF$ means $A$ matches $D$ , $B$ matches $E$ and $C$ matches $F$ . So $AB = DE$ , $\angle B = \angle E$ , and so on.

Worked example: Describing a sequence of rigid motions

Show that triangle $ABC$ with $A(1, 1)$ , $B(3, 1)$ , $C(1, 4)$ is congruent to triangle $A'B'C'$ with $A'(-1, -4)$ , $B'(-3, -4)$ , $C'(-1, -1)$ .

Triangle ABC and triangle A′B′C′.Open in grapher →

In $ABC$ , point $B$ is to the right of $A$ . In the image, $B'$ is to the left of $A'$ , and $C'$ is still above $A'$ . The triangle has been flipped left-to-right, which suggests a reflection across the y-axis.

Reflect across the y-axis: $A(1, 1) \to (-1, 1)$ , $B(3, 1) \to (-3, 1)$ , $C(1, 4) \to (-1, 4)$ .
Translate 5 units down: $(-1, 1) \to (-1, -4)$ , $(-3, 1) \to (-3, -4)$ , $(-1, 4) \to (-1, -1)$ .

That's exactly $A'B'C'$ . A reflection followed by a translation carried $ABC$ onto $A'B'C'$ , so $\triangle ABC \cong \triangle A'B'C'$ .

Similar figures

Definition

Similar

Two figures are similar if one can be carried onto the other by a sequence of rigid motions and dilations. The symbol $\sim$ means "is similar to."

Similar figures have the same shape, but possibly a different size.

Similar figures

Corresponding angles are equal.
Corresponding sides are proportional: every side of one figure is the matching side of the other times the same scale factor $k$ .
Congruent figures are also similar, with $k = 1$ .

Worked example: A dilation and a translation

Triangle $ABC$ has vertices $A(1, 1)$ , $B(3, 1)$ and $C(1, 2)$ . Dilate it by a scale factor of 2 about the origin, then translate the image 8 units left. Are $ABC$ and the final triangle congruent, similar, or neither?

Dilate by 2: $(2, 2)$ , $(6, 2)$ , $(2, 4)$ .
Translate 8 left: $A'(-6, 2)$ , $B'(-2, 2)$ , $C'(-6, 4)$ .

Triangle ABC and its image after a dilation by 2 and a translation.Open in grapher →

The sequence includes a dilation with $k = 2$ , so the triangles are similar but not congruent. Every side of $A'B'C'$ is twice as long as the matching side of $ABC$ : for example, $AB = 2$ and $A'B' = 4$ .

Finding missing sides

If you know two figures are similar, find the scale factor from one pair of matching sides, then use it for the others.

Worked example: Using the scale factor

$\triangle ABC \sim \triangle DEF$ . In $\triangle ABC$ , $AB = 6$ , $BC = 8$ and $AC = 10$ . In $\triangle DEF$ , $DE = 9$ . Find $EF$ and $DF$ .

The letter order says $AB$ matches $DE$ , $BC$ matches $EF$ and $AC$ matches $DF$ .

Scale factor from $ABC$ to $DEF$ : $k = \dfrac{DE}{AB} = \dfrac{9}{6} = 1.5$ .

EF = 8 \times 1.5 = 12 \qquad DF = 10 \times 1.5 = 15

Common mistake

Match sides using the letter order, not by how the figures look on the page. Similar triangles are often drawn turned or flipped, and the side that looks like the "bottom" in one might not match the "bottom" in the other.

Tip

To test whether two triangles could be similar, divide each side of one by the matching side of the other, from shortest to longest. If all the ratios are the same, the sides are proportional.

Practice

Practice 1

Triangle $P$ is rotated $90^\circ$ and then translated 4 units up to make triangle $Q$ . Which is true?

Practice 2

A figure is dilated by a scale factor of 3 and then reflected across the x-axis. How is the image related to the original figure?

Practice 3

$\triangle PQR \cong \triangle XYZ$ . If $PQ = 11$ and $m\angle Q = 48^\circ$ , what is $m\angle Y$ in degrees?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

$\triangle ABC \sim \triangle DEF$ with $AB = 4$ , $BC = 6$ and $DE = 10$ . Find $EF$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

$\triangle JKL \sim \triangle MNO$ . The scale factor from $\triangle JKL$ to $\triangle MNO$ is $\dfrac{1}{3}$ , and $MN = 5$ . Find $JK$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Triangle ABC and triangle A′B′C′.Open in grapher →

Practice 6

Which sequence of transformations carries triangle $ABC$ onto triangle $A'B'C'$ in the graph above?

Practice 7

$\triangle RST \sim \triangle UVW$ with $RS = 8$ and $UV = 6$ . The perimeter of $\triangle RST$ is 30. What is the perimeter of $\triangle UVW$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

One triangle has sides 3, 4 and 5. Another has sides 6, 8 and 11. Are the triangles similar?