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<h2 class="wp-block-heading">Key Concepts</h2>



<ul class="wp-block-list"><li>Understand rotation</li><li>Draw a rotation</li><li>Apply coordinate rules for rotation</li><li>Rotate a figure using coordinate rules</li></ul>



<h2 class="wp-block-heading">Rotation</h2>



<p>Rotation is a transformation in which a figure is turned about a fixed point called the center of rotation. Rays drawn from the center of rotation to a point and their image form the angle of rotation.&nbsp;</p>



<p>A rotation about a point A through an angle of x<sup>0</sup> maps every point B in the plane to a point B’ so that one of the following properties is true.&nbsp;</p>



<ul class="wp-block-list"><li>If B is not the center of rotation A, then BP = B’P and m∠ BPB’= x<sup>0</sup> or&nbsp;</li><li>If B is the center of rotation A, then the image of B is B.&nbsp;</li></ul>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-830.png" alt="Rotation" class="wp-image-15406" width="253" height="229" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-830.png 676w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-830-300x272.png 300w" sizes="auto, (max-width: 253px) 100vw, 253px" /></figure></div>


<h3 class="wp-block-heading">Direction of rotation</h3>



<p>Rotations can be clockwise or counterclockwise.&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-827.png" alt="Direction of rotation clockwise" class="wp-image-15402" width="164" height="176" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-827.png 733w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-827-280x300.png 280w" sizes="auto, (max-width: 164px) 100vw, 164px" /></figure></div>

<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-828.png" alt="Direction of rotation counter clockwise" class="wp-image-15404" width="236" height="177" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-828.png 936w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-828-300x226.png 300w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-828-768x578.png 768w" sizes="auto, (max-width: 236px) 100vw, 236px" /></figure></div>


<p><strong>Note:</strong>&nbsp;</p>



<p>In this chapter, all rotations are counterclockwise.&nbsp;</p><div class="fillWidthInMob"><a loading='lazy' class="showAdInAllRes"  data-toggle="modal" data-target=#homeLeadForm data-heading="Get Expert Guidance" href="#"><img class="img-fluid" alt="parallel" src=https://a.storyblok.com/f/128066/1100x230/00b28b4876/11-applying-for-collages_leaderboard-1100x230.jpg /></a></div>



<h3 class="wp-block-heading">Draw a rotation</h3>



<p>Draw a 120<sup>0</sup> rotation of ΔABC about P.&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-833.png" alt="Draw a 1200 rotation of ΔABC about P" class="wp-image-15409" width="199" height="138"/></figure></div>


<p><strong>Solution: </strong></p>



<p><strong>Step 1</strong>: Draw a segment from A to P. </p>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-829.png" alt="Draw a segment from A to P" class="wp-image-15405" width="412" height="298" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-829.png 652w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-829-300x217.png 300w" sizes="auto, (max-width: 412px) 100vw, 412px" /></figure></div>


<p><strong>Step 2: </strong>Draw a ray to form a 120<sup>0</sup> PA</p>



<p>.&nbsp;</p><div class="fillWidthInMob"><a loading='lazy' class="showAdInAllRes"  data-toggle="modal" data-target=#homeLeadForm data-heading="Get Expert Guidance" href="#"><img class="img-fluid" alt="parallel" src=https://a.storyblok.com/f/128066/1100x230/00b28b4876/11-applying-for-collages_leaderboard-1100x230.jpg /></a></div>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-838.png" alt="Draw a ray to form a 1200 PA" class="wp-image-15414" width="349" height="203"/></figure></div>


<p><strong>Step 3: </strong>Draw A’ so that PA’ = PA </p>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-836.png" alt="Draw A’ so that PA’ = PA " class="wp-image-15412" width="364" height="232"/></figure></div>


<p><strong>Step 4</strong>: Repeat steps 1-3 for each vertex. Draw ΔA′B′C&#8217;</p>



<h3 class="wp-block-heading">Rotations about the origin</h3>



<p>If a rotation is shown in a coordinate plane, the center of rotation is the origin.&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-839.png" alt="Rotations about the origin" class="wp-image-15415" width="338" height="345" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-839.png 633w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-839-294x300.png 294w" sizes="auto, (max-width: 338px) 100vw, 338px" /></figure></div>


<p>The diagram shows rotations of point A 130°, 220°, and 310° about the origin. A rotation of 360° returns a figure to its original coordinates.&nbsp;</p>



<p>There are coordinate rules that can be used to find the coordinates of a point after rotations of 90°, 180°, or 270° about the origin.&nbsp;</p>



<h4 class="wp-block-heading">Coordinate rules for rotations about the origin: </h4>



<p>When a point (a, b) is rotated counterclockwise about the origin, the following are true:&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-835.png" alt="Coordinate rules for rotations about the origin" class="wp-image-15411" width="368" height="376" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-835.png 659w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-835-294x300.png 294w" sizes="auto, (max-width: 368px) 100vw, 368px" /></figure></div>


<ol class="wp-block-list"><li>For a rotation of 90°°, (a, b) → (–b, a). </li><li>For a rotation of 180°°, (a, b) → (–a, –b). </li><li>For a rotation of 270°°, (a, b) → (b, –a). </li></ol>



<p><strong>Example: </strong></p>



<p>When a point (3, 4) is rotated counterclockwise about the origin. Find the coordinate after the rotation of 90°, 180° rotation.&nbsp;</p>



<p><strong>Solution: </strong></p>



<p>For a rotation of 90°, (a, b) → (–b, a).&nbsp;</p>



<p>(3,4) → (–4, 3)&nbsp;</p>



<p>For a rotation of 180°, (a, b) → (–a, –b).&nbsp;</p>



<p>(3,4) → (–3, –4) </p>



<p>For a rotation of 270°, (a, b) → (b, –a). </p>



<p>(3,4) → (4, –3)&nbsp;</p>



<h3 class="wp-block-heading">Rotate a figure using the coordinate rules:&nbsp;</h3>



<p>Now, let us use the coordinate rules to rotate a figure in the coordinate plane.&nbsp;</p>



<p><strong>Example 1: </strong></p>



<p>Graph quadrilateral ABCD with vertices A(3, 1), B(5, 1), C(5, –3), and D(2, –1). Then rotate the quadrilateral 270° about the origin.&nbsp;</p>



<p><strong>Solution:</strong>&nbsp;</p>



<p>Graph ABCD. Use the coordinate rule for a 270° rotation to find the images of the vertices.&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-837.png" alt="Graph ABCD. Use the coordinate rule for a 270° rotation to find the images of the vertices." class="wp-image-15413" width="355" height="271" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-837.png 335w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-837-300x229.png 300w" sizes="auto, (max-width: 355px) 100vw, 355px" /></figure></div>


<p><strong>(a, b)</strong> <strong>→</strong> <strong>(b, –a)</strong>&nbsp;</p>



<p>A(3, 1) <strong>→</strong> A’(1, –3)&nbsp;</p>



<p>B(5, 1) <strong>→</strong> B’(1, –5)&nbsp;</p>



<p>C(5, –3) <strong>→</strong> C’(–3, –5)&nbsp;</p>



<p>D(2, –1) <strong>→</strong> D’(–1, –2)&nbsp;</p>



<p>Now graph the image A’B’C’D’.&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-843.png" alt="Now graph the image A’B’C’D" class="wp-image-15421" width="322" height="273" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-843.png 784w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-843-300x254.png 300w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-843-768x650.png 768w" sizes="auto, (max-width: 322px) 100vw, 322px" /></figure></div>


<p><strong>Example 2:</strong> </p>



<p>Graph a triangle ABC with vertices A(3, 0), B(4, 3), and C(6, 0). Rotate the triangle 90° about the origin.&nbsp;</p>



<p><strong>Solution:</strong>&nbsp;</p>



<p>Graph ABC. Use the coordinate rule for a 90° rotation to find the images of the vertices.&nbsp;</p>



<p><strong>(a, b)</strong> → <strong>(-b, a)</strong>&nbsp;</p>



<p>A(3, 0) → A’(0,3)&nbsp;</p>



<p>B(4, 3) → B’(–3, 4)&nbsp;</p>



<p>C(6, 0) → C’(0, 6)&nbsp;</p>



<p>Now graph the image A’B’C’.&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="370" height="314" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-840.png" alt="Now graph the image A’B’C" class="wp-image-15418" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-840.png 370w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-840-300x255.png 300w" sizes="auto, (max-width: 370px) 100vw, 370px" /></figure></div>


<p><strong>Example 3:</strong> </p>



<p>Graph quadrilateral ABCD with vertices A(3, 1), B(5, 1), C(5, –3), and D(2, –1). Then rotate the quadrilateral 180° about the origin.&nbsp;</p>



<p><strong>Solution:</strong>&nbsp;</p>



<p>Graph ABCD. Use the coordinate rule for a 180° rotation to find the images of the vertices.&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="405" height="284" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-841.png" alt="Graph ABCD. Use the coordinate rule for a 180° rotation to find the images of the vertices. " class="wp-image-15419" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-841.png 405w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-841-300x210.png 300w" sizes="auto, (max-width: 405px) 100vw, 405px" /></figure></div>


<p><strong>(a, b)</strong> → <strong>(–a, –b)</strong>&nbsp;</p>



<p>A(3, 1) → A’(–3, –1)&nbsp;</p>



<p>B(5, 1) → B’(–5, -1)&nbsp;</p>



<p>C(5, –3) → C’(–5, 3)&nbsp;</p>



<p>D(2, –1) → D’(–2, 1)&nbsp;</p>



<p>Now graph the image A’B’C’D’.&nbsp;</p>


<div class="wp-block-image">
<figure class="aligncenter size-full"><img loading="lazy" decoding="async" width="411" height="309" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-842.png" alt="Now graph the image A’B’C’D" class="wp-image-15420" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-842.png 411w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-842-300x226.png 300w" sizes="auto, (max-width: 411px) 100vw, 411px" /></figure></div>


<h3 class="wp-block-heading">Summary </h3>



<ul class="wp-block-list"><li>Rotation is a transformation in which a figure is turned about a fixed point called the center of rotation. Rays drawn from the center of rotation to a point and their image form the angle of rotation. </li><li>Rotations can be clockwise or counterclockwise. </li><li>If a rotation is shown in a coordinate plane, the center of rotation is the origin. </li><li>When a point (a, b) is rotated counterclockwise about the origin, the following are true: <ul><li>For a rotation of 90°,(a, b) + (-b, a). </li><li>For a rotation of 180°,(a, b) + (-a, -b) </li><li>For a rotation of 270°,(a, b) → (b,-a).</li></ul></li></ul>



<h2 class="wp-block-heading">Exercise</h2>



<ul class="wp-block-list"><li>Draw a 90° rotation of AABC about A. AB= 3 cm, BC= 4 cm and AC= 5 cm.</li><li>Draw a 180° rotation of AABC about A. AB= 3 cm, BC=4 cm and AC=5 cm</li><li>When a point (8,-3) is rotated counterclockwise about the origin. Find the coordinate after the rotation of 90°, 1800,006) rotation.</li><li>When a point (-1,-4) is rotated counterclockwise about the origin. Find the coordinate after the rotation of 90°, 180°,006 rotation.</li><li>Trace ADEF and P. Then draw a 50° rotation of ADEF about P.</li></ul>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1463.png" alt="Exercise 5" class="wp-image-16449" width="246" height="132" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1463.png 303w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1463-300x161.png 300w" sizes="auto, (max-width: 246px) 100vw, 246px" /></figure></div>


<ul class="wp-block-list"><li>Graph ARST with vertices R(3,0), S(4,3), and T6,0). Rotate the triangle 90° about the origin.</li><li>Graph ARST with vertices R(3,0), S(4,3), and T6,0). Rotate the triangle 270° about the origin.</li><li>Rotate the figure the given number of degrees about the origin. List the coordinates of the vertices of the image.</li></ul>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1489.png" alt="Exercise 8" class="wp-image-16476" width="239" height="227" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1489.png 403w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1489-300x284.png 300w" sizes="auto, (max-width: 239px) 100vw, 239px" /></figure></div>


<ul class="wp-block-list"><li>Rotate the figure the given number of degrees about the origin. List the coordinates of the vertices of the image.</li></ul>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1490.png" alt="Exercise 9" class="wp-image-16477" width="235" height="213" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1490.png 414w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1490-300x272.png 300w" sizes="auto, (max-width: 235px) 100vw, 235px" /></figure></div>


<ul class="wp-block-list"><li>Rotate the figure the given number of degrees about the origin. List the coordinates of the vertices of the image. </li></ul>


<div class="wp-block-image">
<figure class="aligncenter size-full is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1491.png" alt="Exercise 10" class="wp-image-16478" width="242" height="224" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1491.png 403w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1491-300x278.png 300w" sizes="auto, (max-width: 242px) 100vw, 242px" /></figure></div>


<h3 class="wp-block-heading">Concept Map</h3>


<div class="wp-block-image">
<figure class="aligncenter size-large is-resized"><img loading="lazy" decoding="async" src="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-907-1024x329.png" alt="" class="wp-image-15521" width="686" height="220" srcset="https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-907-1024x329.png 1024w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-907-300x96.png 300w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-907-768x247.png 768w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-907-1536x494.png 1536w, https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-907.png 1561w" sizes="auto, (max-width: 686px) 100vw, 686px" /></figure></div>


<h3 class="wp-block-heading">What we have learnt</h3>



<ul class="wp-block-list"><li>Understand rotation</li><li>Draw a rotation</li><li>Apply coordinate rules for rotation</li><li>Rotate a figure using coordinate rules.</li></ul>



<p>.&nbsp;</p>
