Task Sets: Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, and dilations - Common Core Math - Grade 8

Recall / Warm-Up

These problems review skills you already know: corresponding sides, dilation rules, and the three rigid motions.

$A small triangle ABC beside a larger same-shape triangle DEF.$

Triangle $ABC$ is similar to triangle $DEF$ , written $\triangle ABC \sim \triangle DEF$ . In this statement, which side of $\triangle DEF$ corresponds to side $AB$ of $\triangle ABC$ ?

Side $DE$

Side $EF$

Side $DF$

Side $FE$

A sequence of transformations uses only translations, reflections, and rotations (no dilation). Two figures connected by such a sequence are always which of the following?

Congruent (and therefore also similar)

Similar but never congruent

Neither congruent nor similar

Congruent but not similar

$A coordinate grid with a point at (2, 1) and the origin marked.$

A dilation centered at the origin with scale factor $3$ is applied to the point $(2, 1)$ . What is the x-coordinate of the image point?

Fluency Practice

Find scale factors, check proportionality, and describe transformation sequences. Show your work.

$A coordinate grid showing a small triangle ABC and a larger triangle DEF of the same shape.$

Triangle $ABC$ has vertices $A(1, 1)$ , $B(3, 1)$ , $C(1, 3)$ . Triangle $DEF$ has vertices $D(2, 2)$ , $E(6, 2)$ , $F(2, 6)$ , and $\triangle ABC \sim \triangle DEF$ . Using corresponding sides $AB$ and $DE$ , what is the scale factor from $\triangle ABC$ to $\triangle DEF$ ?

$A small 3 by 2 rectangle beside a larger 9 by 6 rectangle of the same proportions.$

Rectangle $ABCD$ is $3$ units wide and $2$ units tall. Rectangle $EFGH$ is $9$ units wide and $6$ units tall, and the two rectangles are similar. What is the common ratio of each side of $EFGH$ to its corresponding side of $ABCD$ (the scale factor)?

$A coordinate grid showing triangle PQR near the origin and a larger triangle STU shifted up and to the right.$

Triangle $PQR$ has vertices $P(1, 0)$ , $Q(3, 0)$ , $R(1, 2)$ . Triangle $STU$ has vertices $S(5, 3)$ , $T(9, 3)$ , $U(5, 7)$ , and the triangles are similar with scale factor $2$ . Which sequence maps $\triangle PQR$ onto $\triangle STU$ ?

Dilate by scale factor $2$ centered at the origin, then translate $3$ units right and $3$ units up.

Translate $3$ units right and $3$ units up only (no dilation).

Dilate by scale factor $2$ centered at the origin only (no translation).

Reflect across the y-axis, then dilate by scale factor $2$ .

$A coordinate grid showing triangle ABC in the first quadrant and a larger triangle DEF in the third quadrant.$

Triangle $ABC$ has vertices $A(0, 0)$ , $B(2, 0)$ , $C(0, 3)$ . Triangle $DEF$ has vertices $D(0, 0)$ , $E(-4, 0)$ , $F(0, -6)$ , and the triangles are similar with scale factor $2$ . Which sequence maps $\triangle ABC$ onto $\triangle DEF$ ?

Dilate by scale factor $2$ centered at the origin, then rotate $180\degree$ about the origin.

Dilate by scale factor $2$ centered at the origin only (no rotation).

Rotate $180\degree$ about the origin only (no dilation).

Translate down and left only.

Two similar triangles have corresponding sides in the ratio $\frac{15}{6}$ . A student must report the scale factor as a fully reduced fraction. What is the scale factor? Write it as a fraction.

Exercises: Understand Similarity Through Sequences of Transformations

Recall / Warm-Up

Fluency Practice