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Compositions and Congruence: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered

Question 1 of 12

1

First reflect $P(2,-4)$ across the $y$ -axis, then translate by $(a,b)$ to reach $P''(5,1)$ . What translation is required?

Answer choices for question 1 A $(a,b)=(3,5)$ B $(a,b)=(-7,-5)$ C $(a,b)=(7,-3)$ D $(a,b)=(7,5)$
2

Compare translation by $(a,b)$ and reflection across the $x$ -axis in both orders. For which condition do the orders agree for every point?

Answer choices for question 2 A $a=0$ , with any real $b$ B $b=0$ , with any real $a$ C $a=b$ , with any real value D $b=1$ , with any real $a$
3

A sequence first translates by $(2,-1)$ , then reflects across $y=x$ , then makes a half-turn about the origin. Which ordered sequence is its inverse?

Answer choices for question 3 A Translate by $(-2,1)$ , reflect across $y=x$ , half-turn about the origin B Half-turn about the origin, reflect across $y=-x$ , translate by $(2,-1)$ C Half-turn about the origin, reflect across $y=x$ , translate by $(-2,1)$ D Translate by $(2,-1)$ , reflect across $y=x$ , half-turn about the origin
4

Mirrors m and n meet at O, with the chosen angle from m to n equal to 62 degrees counterclockwise. What results from first reflecting in n, then in m?

Answer choices for question 4 A A 124-degree clockwise rotation about O B A 124-degree counterclockwise rotation about O C A 62-degree clockwise rotation about O D A 248-degree clockwise rotation about O
5

First reflect $H(3,-8)$ across $y=x$ , then across $y=-x$ . Which statement correctly compares this with the reversed order?

Answer choices for question 5 A Both give $H''(3,-8)$ , the starting point. B Both give $H''(-3,8)$ , a half-turn image. C The reversed order gives $H''(8,-3)$ instead. D Both give $H''(-8,3)$ , a quarter-turn image.
6

A mapping preserves the length of the vertical segment from $(0,1)$ to $(0,4)$ . Someone concludes that this alone makes the mapping rigid. Which counterexample settles the claim?

Answer choices for question 6 A $(x,y)\mapsto(x+2,y)$ preserves every distance. B $(x,y)\mapsto(-x,y)$ fixes the given segment. C $(x,y)\mapsto(-x,-y)$ preserves the given length. D $(x,y)\mapsto(2x,y)$ preserves that segment but doubles the length from $(0,0)$ to $(1,0)$ .
7

A specified rigid sequence sends $A$ to $R$ , $B$ to $T$ , and $C$ to $S$ . Which side and angle correspond to $AC$ and $\angle ABC$ ?

Answer choices for question 7 A $RS$ and $\angle RTS$ B $RT$ and $\angle RST$ C $TS$ and $\angle TRS$ D $RS$ and $\angle RST$
8

A glide first reflects across a horizontal mirror $y=k$ , then translates four units right. It maps $P(1,7)$ to $P''(5,-1)$ . Find $k$ .

Answer choices for question 8 A $-3$ B $4$ C $3$ D $-4$
9

Start at $W(-4,1)$ . First reflect across $y=-x$ , then rotate $90^\circ$ clockwise about the origin, then translate three units down. What is the final image?

Answer choices for question 9 A $(4,-2)$ B $(-4,-2)$ C $(4,4)$ D $(-1,1)$
10

Two reflections followed by a translation map an angle measuring $(7t-9)^\circ$ to its corresponding angle measuring $(4t+36)^\circ$ . Find their common measure.

Answer choices for question 10 A $15^\circ$ B $45^\circ$ C $96^\circ$ D $105^\circ$
11

Sequence A rotates 90 degrees counterclockwise about the origin, then translates two units right. Sequence B reverses those two motions. Where do they send the origin?

Answer choices for question 11 A Both send it to (2,0). B A sends it to (2,0); B sends it to (0,2). C A sends it to (0,2); B sends it to (2,0). D Both send it to (0,2).
12

Triangle $ABC$ has $A(1,0)$ , $B(3,0)$ , $C(1,2)$ . Target $DEF$ has $D(0,-1)$ , $E(0,-3)$ , $F(-2,-1)$ . Which sequence maps $A,B,C$ to $D,E,F$ respectively?

Answer choices for question 12 A Rotate 90 degrees clockwise about the origin, then reflect across the x-axis. B Reflect across the y-axis, then rotate 90 degrees clockwise about the origin. C Translate one unit left, then rotate 90 degrees clockwise about the origin. D Reflect across the x-axis, then rotate 90 degrees clockwise about the origin.

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