Question

complete all problems below. show all work. *note: images are not drawn to scale. 1.) perform the transformations. a.) translate δkjl 4 units left and 1 unit up. label with prime notation. b.) rotate δqpr 270° counterclockwise about the origin. label with prime notation. c.) reflect qrsp across the x-axis. label with prime notation. d.) rotate δlmk 90° clockwise about the origin. label with prime notation.

Explanation:

Part (a): Translate $\boldsymbol{\triangle KJL}$ 4 units left and 1 unit up

Step 1: Identify coordinates of $\boldsymbol{\triangle KJL}$

• Let’s assume the grid has integer coordinates. From the graph:
• $J$: $(1, -1)$ (since it’s 1 unit right on x, 1 unit down on y from origin)
• $K$: $(2, 3)$ (2 units right, 3 units up)
• $L$: $(3, -1)$ (3 units right, 1 unit down)

Step 2: Apply translation rule

Translation 4 units left (subtract 4 from $x$) and 1 unit up (add 1 to $y$):

• For a point $(x, y)$, new coordinates: $(x - 4, y + 1)$

• $J'$: $(1 - 4, -1 + 1) = (-3, 0)$
• $K'$: $(2 - 4, 3 + 1) = (-2, 4)$
• $L'$: $(3 - 4, -1 + 1) = (-1, 0)$

Step 3: Plot and label

Plot $J'(-3, 0)$, $K'(-2, 4)$, $L'(-1, 0)$ and connect to form $\triangle K'J'L'$.

Part (b): Rotate $\boldsymbol{\triangle QPR}$ $270^\circ$ counterclockwise about the origin

Step 1: Identify coordinates of $\boldsymbol{\triangle QPR}$

From the graph:

• $P$: $(2, -1)$
• $Q$: $(3, 1)$
• $R$: $(4, -2)$

Step 2: Apply $270^\circ$ counterclockwise rotation rule

Rotation rule: $(x, y) \to (y, -x)$ (for $270^\circ$ counterclockwise about origin)

• $P'$: $(-1, -2)$ (since $x=2, y=-1$ → $(y, -x) = (-1, -2)$)
• $Q'$: $(1, -3)$ (since $x=3, y=1$ → $(y, -x) = (1, -3)$)
• $R'$: $(-2, -4)$ (since $x=4, y=-2$ → $(y, -x) = (-2, -4)$)

Step 3: Plot and label

Plot $P'(-1, -2)$, $Q'(1, -3)$, $R'(-2, -4)$ and connect to form $\triangle Q'P'R'$.

Part (c): Reflect $\boldsymbol{QRSP}$ across the $\boldsymbol{x}$-axis

Step 1: Identify coordinates of $\boldsymbol{QRSP}$

From the graph:

• $Q$: $(2, 2)$
• $R$: $(3, 2)$
• $S$: $(4, 0)$
• $P$: $(2, -3)$

Step 2: Apply reflection rule across $x$-axis

Reflection rule: $(x, y) \to (x, -y)$

• $Q'$: $(2, -2)$ (since $y=2$ → $-y = -2$)
• $R'$: $(3, -2)$ (since $y=2$ → $-y = -2$)
• $S'$: $(4, 0)$ (since $y=0$ → $-y = 0$)
• $P'$: $(2, 3)$ (since $y=-3$ → $-y = 3$)

Step 3: Plot and label

Plot $Q'(2, -2)$, $R'(3, -2)$, $S'(4, 0)$, $P'(2, 3)$ and connect to form $Q'R'S'P'$.

Part (d): Rotate $\boldsymbol{\triangle LMK}$ $90^\circ$ clockwise about the origin

Step 1: Identify coordinates of $\boldsymbol{\triangle LMK}$

From the graph:

• $L$: $(4, -1)$
• $M$: $(5, -1)$
• $K$: $(4, -3)$

Step 2: Apply $90^\circ$ clockwise rotation rule

Rotation rule: $(x, y) \to (y, -x)$ (for $90^\circ$ clockwise about origin)

• $L'$: $(-1, -4)$ (since $x=4, y=-1$ → $(y, -x) = (-1, -4)$)
• $M'$: $(-1, -5)$ (since $x=5, y=-1$ → $(y, -x) = (-1, -5)$)
• $K'$: $(-3, -4)$ (since $x=4, y=-3$ → $(y, -x) = (-3, -4)$)

Step 3: Plot and label

Plot $L'(-1, -4)$, $M'(-1, -5)$, $K'(-3, -4)$ and connect to form $\triangle L'M'K'$.

Final Answer (Summary of Transformations)

• (a) $\triangle K'J'L'$ with vertices $(-3, 0)$, $(-2, 4)$, $(-1, 0)$.
• (b) $\triangle Q'P'R'$ with vertices $(-1, -2)$, $(1, -3)$, $(-2, -4)$.
• (c) $Q'R'S'P'$ with vertices $(2, -2)$, $(3, -2)$, $(4, 0)$, $(2, 3)$.
• (d) $\triangle L'M'K'$ with vertices $(-1, -4)$, $(-1, -5)$, $(-3, -4)$.

(Note: For precise plotting, use the calculated coordinates on the grid.)

Answer:

Part (a): Translate $\boldsymbol{\triangle KJL}$ 4 units left and 1 unit up

Step 1: Identify coordinates of $\boldsymbol{\triangle KJL}$

• Let’s assume the grid has integer coordinates. From the graph:
• $J$: $(1, -1)$ (since it’s 1 unit right on x, 1 unit down on y from origin)
• $K$: $(2, 3)$ (2 units right, 3 units up)
• $L$: $(3, -1)$ (3 units right, 1 unit down)

Step 2: Apply translation rule

Translation 4 units left (subtract 4 from $x$) and 1 unit up (add 1 to $y$):

• For a point $(x, y)$, new coordinates: $(x - 4, y + 1)$

• $J'$: $(1 - 4, -1 + 1) = (-3, 0)$
• $K'$: $(2 - 4, 3 + 1) = (-2, 4)$
• $L'$: $(3 - 4, -1 + 1) = (-1, 0)$

Step 3: Plot and label

Plot $J'(-3, 0)$, $K'(-2, 4)$, $L'(-1, 0)$ and connect to form $\triangle K'J'L'$.

Part (b): Rotate $\boldsymbol{\triangle QPR}$ $270^\circ$ counterclockwise about the origin

Step 1: Identify coordinates of $\boldsymbol{\triangle QPR}$

From the graph:

• $P$: $(2, -1)$
• $Q$: $(3, 1)$
• $R$: $(4, -2)$

Step 2: Apply $270^\circ$ counterclockwise rotation rule

Rotation rule: $(x, y) \to (y, -x)$ (for $270^\circ$ counterclockwise about origin)

• $P'$: $(-1, -2)$ (since $x=2, y=-1$ → $(y, -x) = (-1, -2)$)
• $Q'$: $(1, -3)$ (since $x=3, y=1$ → $(y, -x) = (1, -3)$)
• $R'$: $(-2, -4)$ (since $x=4, y=-2$ → $(y, -x) = (-2, -4)$)

Step 3: Plot and label

Plot $P'(-1, -2)$, $Q'(1, -3)$, $R'(-2, -4)$ and connect to form $\triangle Q'P'R'$.

Part (c): Reflect $\boldsymbol{QRSP}$ across the $\boldsymbol{x}$-axis

Step 1: Identify coordinates of $\boldsymbol{QRSP}$

From the graph:

• $Q$: $(2, 2)$
• $R$: $(3, 2)$
• $S$: $(4, 0)$
• $P$: $(2, -3)$

Step 2: Apply reflection rule across $x$-axis

Reflection rule: $(x, y) \to (x, -y)$

• $Q'$: $(2, -2)$ (since $y=2$ → $-y = -2$)
• $R'$: $(3, -2)$ (since $y=2$ → $-y = -2$)
• $S'$: $(4, 0)$ (since $y=0$ → $-y = 0$)
• $P'$: $(2, 3)$ (since $y=-3$ → $-y = 3$)

Step 3: Plot and label

Plot $Q'(2, -2)$, $R'(3, -2)$, $S'(4, 0)$, $P'(2, 3)$ and connect to form $Q'R'S'P'$.

Part (d): Rotate $\boldsymbol{\triangle LMK}$ $90^\circ$ clockwise about the origin

Step 1: Identify coordinates of $\boldsymbol{\triangle LMK}$

From the graph:

• $L$: $(4, -1)$
• $M$: $(5, -1)$
• $K$: $(4, -3)$

Step 2: Apply $90^\circ$ clockwise rotation rule

Rotation rule: $(x, y) \to (y, -x)$ (for $90^\circ$ clockwise about origin)

• $L'$: $(-1, -4)$ (since $x=4, y=-1$ → $(y, -x) = (-1, -4)$)
• $M'$: $(-1, -5)$ (since $x=5, y=-1$ → $(y, -x) = (-1, -5)$)
• $K'$: $(-3, -4)$ (since $x=4, y=-3$ → $(y, -x) = (-3, -4)$)

Step 3: Plot and label

Plot $L'(-1, -4)$, $M'(-1, -5)$, $K'(-3, -4)$ and connect to form $\triangle L'M'K'$.

Final Answer (Summary of Transformations)

• (a) $\triangle K'J'L'$ with vertices $(-3, 0)$, $(-2, 4)$, $(-1, 0)$.
• (b) $\triangle Q'P'R'$ with vertices $(-1, -2)$, $(1, -3)$, $(-2, -4)$.
• (c) $Q'R'S'P'$ with vertices $(2, -2)$, $(3, -2)$, $(4, 0)$, $(2, 3)$.
• (d) $\triangle L'M'K'$ with vertices $(-1, -4)$, $(-1, -5)$, $(-3, -4)$.

(Note: For precise plotting, use the calculated coordinates on the grid.)