Exercises: Draw transformed figures - Common Core Math - HS Geometry

A

Warm-Up: Review What You Know

These problems review skills you have already learned.

1.

Point $P$ has coordinates $(3, -2)$ . Under a translation by vector $(4, 5)$ , what are the coordinates of the image $P'$ ?

A.

$(7, 3)$

B.

$(-1, -7)$

C.

$(7, -7)$

D.

$(3, 3)$

2.

Point $Q$ has coordinates $(-5, 2)$ . What are the coordinates of its image after a reflection over the $y$ -axis?

A.

$(-5, -2)$

B.

$(5, 2)$

C.

$(2, -5)$

D.

$(-2, 5)$

3.

Point $R$ has coordinates $(4, 1)$ . What are the coordinates of its image after a $90^\circ$ counterclockwise rotation about the origin?

A.

$(-1, 4)$

B.

$(1, -4)$

C.

$(-4, -1)$

D.

$(4, -1)$

B

Fluency Practice

Apply the given transformation to each vertex. List all image coordinates.

1.

Triangle $ABC$ has vertices $A(1, 2)$ , $B(4, 1)$ , and $C(3, 5)$ . It is translated by vector $(3, -2)$ .

Which set of coordinates gives the image $A'B'C'$ ?

A.

$A'(4, 0)$ , $B'(7, -1)$ , $C'(6, 3)$

B.

$A'(4, 0)$ , $B'(7, -1)$ , $C'(3, 5)$

C.

$A'(-2, 4)$ , $B'(1, 3)$ , $C'(0, 7)$

D.

$A'(4, 4)$ , $B'(7, 3)$ , $C'(6, 7)$

2.

Quadrilateral $PQRS$ has vertices $P(0, 0)$ , $Q(4, 0)$ , $R(4, 3)$ , and $S(0, 3)$ . It is translated by vector $(-3, 2)$ .

What are the coordinates of $P'$ , $Q'$ , $R'$ , and $S'$ ?

A.

$P'(-3, 2)$ , $Q'(1, 2)$ , $R'(1, 5)$ , $S'(-3, 5)$

B.

$P'(3, -2)$ , $Q'(7, -2)$ , $R'(7, 1)$ , $S'(3, 1)$

C.

$P'(-3, 2)$ , $Q'(1, 2)$ , $R'(4, 3)$ , $S'(0, 3)$

D.

$P'(-3, 2)$ , $Q'(1, 2)$ , $R'(1, 3)$ , $S'(-3, 3)$

3.

Triangle $DEF$ has vertices $D(-3, 1)$ , $E(0, 4)$ , and $F(2, -1)$ . It is reflected over the $y$ -axis.

What are the coordinates of the image $D'E'F'$ ?

A.

$D'(3, 1)$ , $E'(0, 4)$ , $F'(-2, -1)$

B.

$D'(-3, -1)$ , $E'(0, -4)$ , $F'(2, 1)$

C.

$D'(1, -3)$ , $E'(4, 0)$ , $F'(-1, 2)$

D.

$D'(3, 1)$ , $E'(0, 4)$ , $F'(2, -1)$

4.

Triangle $GHK$ has vertices $G(1, 3)$ , $H(4, 3)$ , and $K(1, 7)$ . It is reflected over the horizontal line $y = 1$ .

What are the coordinates of the image $G'H'K'$ ?

A.

$G'(1, -1)$ , $H'(4, -1)$ , $K'(1, -5)$

B.

$G'(-1, 3)$ , $H'(-4, 3)$ , $K'(-1, 7)$

C.

$G'(1, 3)$ , $H'(4, 3)$ , $K'(1, 7)$

D.

$G'(1, 1)$ , $H'(4, 1)$ , $K'(1, -5)$

5.

Triangle $LMN$ has vertices $L(2, 0)$ , $M(5, 0)$ , and $N(2, 4)$ . It is rotated $90^\circ$ counterclockwise about the origin.

What are the coordinates of the image $L'M'N'$ ?

A.

$L'(0, 2)$ , $M'(0, 5)$ , $N'(-4, 2)$

B.

$L'(0, -2)$ , $M'(0, -5)$ , $N'(4, -2)$

C.

$L'(-2, 0)$ , $M'(-5, 0)$ , $N'(-2, -4)$

D.

$L'(0, 2)$ , $M'(0, 5)$ , $N'(4, -2)$

C

Mixed Practice

These problems test the same skills in different ways.

D

Word Problems

Read each problem carefully and apply the correct transformation.

1.

An artist is creating a repeating tile pattern. She starts with rectangle $ABCD$ with vertices $A(0, 0)$ , $B(3, 0)$ , $C(3, 2)$ , and $D(0, 2)$ , then reflects it over the line $x = 3$ to create a second tile.

What are the coordinates of the image vertex $A'$ ?

A.

$A'(6, 0)$

B.

$A'(0, 0)$

C.

$A'(-6, 0)$

D.

$A'(3, 0)$

2.

A robot arm starts at point $T(5, 2)$ and rotates $90^\circ$ counterclockwise about the pivot point $O(2, 2)$ .

What are the coordinates of the endpoint's new position $T'$ ?

A.

$T'(2, 5)$

B.

$T'(-1, 5)$

C.

$T'(2, -1)$

D.

$T'(5, 2)$

3.

Triangle $PQR$ has vertices $P(1, 1)$ , $Q(4, 1)$ , and $R(1, 3)$ . Triangle $P''Q''R''$ has vertices $P''(-1, -2)$ , $Q''(-4, -2)$ , and $R''(-1, -4)$ .

Specify a sequence of two transformations that maps triangle $PQR$ onto triangle $P''Q''R''$ . Then verify your sequence is correct by computing the image of every vertex.

4.

A graphic designer applies two transformations to a logo shape. She reflects triangle $ABC$ with $A(2,0)$ , $B(5,0)$ , $C(2,3)$ over the $y$ -axis, then translates by $(4, 0)$ . Her assistant instead first translates by $(4, 0)$ , then reflects over the $y$ -axis.

Compute the final image vertices for both orders and explain why the results differ.

E

Find the Mistake

Each problem shows student work with an error. Identify and explain the mistake.

1.

Jordan was asked to translate triangle $ABC$ with $A(1,3)$ , $B(4,3)$ , $C(4,6)$ by vector $(2, -1)$ . Jordan wrote:

$A'(3, 2)$ , $B'(4, 3)$ , $C'(4, 6)$

What mistake did Jordan make?

A.

Jordan applied the translation to vertex $A$ only; vertices $B$ and $C$ were left unchanged.

B.

Jordan added the vector incorrectly — the $x$ -value should decrease, not increase.

C.

Jordan reflected the triangle instead of translating it.

D.

Jordan translated all vertices but used the wrong vector.

2.

Sam was asked to rotate point $V(4, 1)$ by $90^\circ$ counterclockwise about the origin. Sam wrote:

"The rule for $90^\circ$ CCW is $(x,y)\to(y,-x)$ , so $V' = (1, -4)$ ."

What mistake did Sam make?

A.

Sam used the rule for $90^\circ$ clockwise rotation instead of counterclockwise.

B.

Sam used the correct rule but made an arithmetic error.

C.

Sam should have used the $180^\circ$ rotation rule $(x,y)\to(-x,-y)$ .

D.

The rotation was applied to the wrong center point.

F

Challenge Problems

These problems require multi-step reasoning. Show all intermediate steps.

1.

Triangle $FGH$ has vertices $F(0, 0)$ , $G(3, 0)$ , and $H(0, 4)$ .

Triangle $F''G''H''$ has vertices $F''(1, -2)$ , $G''(-2, -2)$ , and $H''(1, 2)$ .

Find a sequence of two transformations that maps $FGH$ onto $F''G''H''$ . Verify your answer by computing all three image vertices.

2.

Triangle $JKL$ has vertices $J(1, 1)$ , $K(3, 1)$ , and $L(1, 3)$ .

A student proposes: rotate $90^\circ$ counterclockwise about the origin, then reflect over the $x$ -axis.