Question

the coordinates of the vertices of △ jkl are j(1, 4) , k(6, 4) , and l(1, 1) .
the coordinates of the vertices of △ jkl are j(0, −4) , k(−5, −4) , and l(0, −1) .

what is the sequence of transformations that maps △ jkl to △ jkl ?

drag and drop the answers into the boxes to correctly complete the statement.

a sequence of transformations that maps △ jkl to △ jkl is a
first transformation followed by a
second transformation

options:
reflection across the x-axis
rotation of 180° about the origin
rotation of 90° counterclockwise about the origin
translation 1 unit left

Explanation:

Step1: Analyze translation

First, check translation. For point \( J(1,4) \), if we translate 1 unit left, we get \( (1 - 1,4)=(0,4) \).

Step2: Analyze reflection across x - axis

Then, reflecting \( (0,4) \) across the x - axis (the rule for reflection across x - axis is \( (x,y)\to(x, - y) \)) gives \( (0,-4) \), which is \( J' \). Let's check \( K(6,4) \): translate 1 unit left: \( (6 - 1,4)=(5,4) \), reflect across x - axis: \( (5,-4) \)? Wait, no, \( K' \) is \( (-5,-4) \). Wait, maybe first translation 1 unit left, then rotation 180°? Wait, rotation 180° rule is \( (x,y)\to(-x,-y) \). Let's try translation 1 unit left first: \( J(1,4)\to(0,4) \), \( K(6,4)\to(5,4) \), \( L(1,1)\to(0,1) \). Then rotation 180°: \( (0,4)\to(0,-4) \), \( (5,4)\to(-5,-4) \), \( (0,1)\to(0,-1) \), which matches \( J' \), \( K' \), \( L' \). Alternatively, let's check reflection and then translation. But the first step: translation 1 unit left, then rotation 180° about the origin. Wait, let's verify the options. The options are reflection across x - axis, rotation 180° about origin, rotation 90° counterclockwise, translation 1 unit left.

Wait, let's re - evaluate. Let's take \( J(1,4) \), \( J'(0,-4) \). Let's see the transformation from \( (1,4) \) to \( (0,-4) \). If we first translate 1 unit left: \( (1 - 1,4)=(0,4) \), then reflect across x - axis: \( (0,-4) \). Wait, but \( K(6,4) \) translated 1 unit left is \( (5,4) \), reflected across x - axis is \( (5,-4) \), but \( K' \) is \( (-5,-4) \). Oh, I made a mistake. So maybe first rotation 180°? Wait, rotation 180° of \( J(1,4) \) is \( (-1,-4) \), not \( (0,-4) \). Wait, translation 1 unit left: \( (1 - 1,4)=(0,4) \), then rotation 180°: \( (0,-4) \), \( (5,4)\to(-5,-4) \), \( (0,1)\to(0,-1) \). Yes! Because translation 1 unit left: \( (x,y)\to(x - 1,y) \), then rotation 180°: \( (x,y)\to(-x,-y) \). So for \( J(1,4) \): translate 1 left: \( (0,4) \), rotate 180°: \( (0,-4) \) (matches \( J' \)). For \( K(6,4) \): translate 1 left: \( (5,4) \), rotate 180°: \( (-5,-4) \) (matches \( K' \)). For \( L(1,1) \): translate 1 left: \( (0,1) \), rotate 180°: \( (0,-1) \) (matches \( L' \)). Alternatively, if we do rotation first, rotation 180° of \( J(1,4) \) is \( (-1,-4) \), then translation 1 unit right would be \( (0,-4) \), but translation 1 unit right is not an option. So the correct sequence is translation 1 unit left followed by rotation 180° about the origin. Wait, but let's check the other way. Wait, the options are: reflection across x - axis, rotation 180° about origin, rotation 90° counterclockwise, translation 1 unit left.

Wait, let's check reflection across x - axis first. Reflect \( J(1,4) \) across x - axis: \( (1,-4) \), then translation? No, \( J' \) is \( (0,-4) \). So translation 1 unit left after reflection? Reflect \( J(1,4) \) to \( (1,-4) \), translate 1 left: \( (0,-4) \). Reflect \( K(6,4) \) to \( (6,-4) \), translate 1 left: \( (5,-4) \), but \( K' \) is \( (-5,-4) \). So that doesn't work. So the correct sequence is translation 1 unit left followed by rotation 180° about the origin.

Answer:

A sequence of transformations that maps \( \triangle JKL \) to \( \triangle J'K'L' \) is a \(\boldsymbol{\text{translation 1 unit left}}\) followed by a \(\boldsymbol{\text{rotation of } 180^\circ \text{ about the origin}}\)