A sequence of translations maps $$\Delta {G} {H} {I}$$ to $$\Delta G^{\prime} H^{\prime} I^{\prime}$$.

- $$\Delta {G} {H} {I}$$ has vertices at $${G}(-8,2), {H}(13,2)$$, and $${I}(-2,10)$$.

- The coordinates of $$G^{\prime}$$ are $$(-1,-3)$$.

What are the coordinates for $$H^{/}$$?


<div id="id_cont_display" class="markdown-body editormd-html-preview"><textarea>A sequence of translations maps $$\Delta {G} {H} {I}$$ to $$\Delta G^{\prime} H^{\prime} I^{\prime}$$.

- $$\Delta {G} {H} {I}$$ has vertices at $${G}(-8,2), {H}(13,2)$$, and $${I}(-2,10)$$.

- The coordinates of $$G^{\prime}$$ are $$(-1,-3)$$.

What are the coordinates for $$H^{/}$$?
</textarea>A sequence of translations maps <math><semantics><mrow><mi mathvariant="normal">Δ</mi><mrow><mi>G</mi></mrow><mrow><mi>H</mi></mrow><mrow><mi>I</mi></mrow></mrow><annotation encoding="application/x-tex">\Delta {G} {H} {I}</annotation></semantics></math>ΔGHI to <math><semantics><mrow><mi mathvariant="normal">Δ</mi><msup><mi>G</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><msup><mi>H</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><msup><mi>I</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\Delta G^{\prime} H^{\prime} I^{\prime}</annotation></semantics></math>ΔG​′​​H​′​​I​′​​.
<ul>
<li><math><semantics><mrow><mi mathvariant="normal">Δ</mi><mrow><mi>G</mi></mrow><mrow><mi>H</mi></mrow><mrow><mi>I</mi></mrow></mrow><annotation encoding="application/x-tex">\Delta {G} {H} {I}</annotation></semantics></math>ΔGHI has vertices at <math><semantics><mrow><mrow><mi>G</mi></mrow><mo>(</mo><mo>−</mo><mn>8</mn><mo separator="true">,</mo><mn>2</mn><mo>)</mo><mo separator="true">,</mo><mrow><mi>H</mi></mrow><mo>(</mo><mn>1</mn><mn>3</mn><mo separator="true">,</mo><mn>2</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">{G}(-8,2), {H}(13,2)</annotation></semantics></math>G(−8,2),H(13,2), and <math><semantics><mrow><mrow><mi>I</mi></mrow><mo>(</mo><mo>−</mo><mn>2</mn><mo separator="true">,</mo><mn>1</mn><mn>0</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">{I}(-2,10)</annotation></semantics></math>I(−2,10).
</li><li>The coordinates of <math><semantics><mrow><msup><mi>G</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup></mrow><annotation encoding="application/x-tex">G^{\prime}</annotation></semantics></math>G​′​​ are <math><semantics><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>3</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(-1,-3)</annotation></semantics></math>(−1,−3).
</li></ul>
What are the coordinates for <math><semantics><mrow><msup><mi>H</mi><mrow><mi mathvariant="normal">/</mi></mrow></msup></mrow><annotation encoding="application/x-tex">H^{/}</annotation></semantics></math>H​/​​?
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<div id="id_solution_display" class="markdown-body editormd-html-preview"><textarea>Given that, $${G}(-8,2)$$ has been translated to $$G^{\prime}(-1,-3)$$

The transformation rule is $$({x}, {y}) \rightarrow(x+7, y-5)$$

The co-ordinate of $$H^{\prime}$$ is $$(13+7,2-5)=(20,-3)$$

The co-ordinate of $$I ^{\prime}$$ is $$(-2+7,10-5)=(5,5)$$ </textarea>Given that, <math><semantics><mrow><mrow><mi>G</mi></mrow><mo>(</mo><mo>−</mo><mn>8</mn><mo separator="true">,</mo><mn>2</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">{G}(-8,2)</annotation></semantics></math>G(−8,2) has been translated to <math><semantics><mrow><msup><mi>G</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup><mo>(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>3</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">G^{\prime}(-1,-3)</annotation></semantics></math>G​′​​(−1,−3)
The transformation rule is <math><semantics><mrow><mo>(</mo><mrow><mi>x</mi></mrow><mo separator="true">,</mo><mrow><mi>y</mi></mrow><mo>)</mo><mo>→</mo><mo>(</mo><mi>x</mi><mo>+</mo><mn>7</mn><mo separator="true">,</mo><mi>y</mi><mo>−</mo><mn>5</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">({x}, {y}) \rightarrow(x+7, y-5)</annotation></semantics></math>(x,y)→(x+7,y−5)
The co-ordinate of <math><semantics><mrow><msup><mi>H</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup></mrow><annotation encoding="application/x-tex">H^{\prime}</annotation></semantics></math>H​′​​ is <math><semantics><mrow><mo>(</mo><mn>1</mn><mn>3</mn><mo>+</mo><mn>7</mn><mo separator="true">,</mo><mn>2</mn><mo>−</mo><mn>5</mn><mo>)</mo><mo>=</mo><mo>(</mo><mn>2</mn><mn>0</mn><mo separator="true">,</mo><mo>−</mo><mn>3</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(13+7,2-5)=(20,-3)</annotation></semantics></math>(13+7,2−5)=(20,−3)
The co-ordinate of <math><semantics><mrow><msup><mi>I</mi><mrow><mi mathvariant="normal">′</mi></mrow></msup></mrow><annotation encoding="application/x-tex">I ^{\prime}</annotation></semantics></math>I​′​​ is <math><semantics><mrow><mo>(</mo><mo>−</mo><mn>2</mn><mo>+</mo><mn>7</mn><mo separator="true">,</mo><mn>1</mn><mn>0</mn><mo>−</mo><mn>5</mn><mo>)</mo><mo>=</mo><mo>(</mo><mn>5</mn><mo separator="true">,</mo><mn>5</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">(-2+7,10-5)=(5,5)</annotation></semantics></math>(−2+7,10−5)=(5,5) 
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