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The triangle π΄π΅πΆ has been transformed onto triangle π΄ prime π΅ prime πΆ prime which has been transformed onto triangle π΄ double prime π΅ double prime πΆ double prime.
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Describe the single transformation that maps π΄π΅πΆ to π΄ prime π΅ prime πΆ prime.
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Describe the single transformation that maps π΄ prime π΅ prime πΆ prime to π΄ double prime π΅ double prime πΆ double prime.
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Hence, are triangles π΄π΅πΆ and π΄ double prime π΅ double prime πΆ double prime similar?
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In this question, we have a series or combinations of transformations which begin with triangle π΄π΅πΆ.
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The first transformation takes us to the second smaller triangle.
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And the second transformation takes us to the larger triangle of π΄ double prime π΅ double prime πΆ double prime.
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We can begin by finding the first transformation between the two smaller triangles.
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We can recall that the four types of transformation are translation, reflection, rotation, and dilation.
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If we look at the two triangles π΄π΅πΆ and π΄ prime π΅ prime πΆ prime, we can see that theyβre the same size.
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Therefore, this is unlikely to be a dilation, as this usually changes the size.
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We can see that our two triangles are at different orientation.
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So, we can rule out translation as this moves the shape but keeps it the same way up.
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The two triangles are not a mirror image of each other.
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So, we can rule out reflection.
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Letβs see if we could describe this transformation as a rotation.
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Starting with triangle π΄π΅πΆ, if we rotated this in a clockwise direction, we could then work out the angles for which this must be rotated.
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Between π΄ and π΄ prime, thereβs a right angle of 90 degrees.
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Between πΆ and πΆ prime, we can also see a 90-degree angle.
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This will confirm that we have a rotation of 90 degrees.
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Notice that weβve found this rotation by moving our vertices through the same point or coordinate.
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This will be the center of rotation.
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So, in order to fully describe this transformation, we need to put together the facts that weβve discovered β the center of rotation, the angle, and the direction.
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We can then give our answer to the first part as a 90-degrees clockwise rotation about the origin.
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We could, of course, also have described this as a 270-degree counterclockwise rotation about the origin.
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Giving the coordinate zero, zero instead of the origin would also have been valid.
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Letβs look at the second question.
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Describe the single transformation that maps π΄ prime π΅ prime πΆ prime to π΄ double prime π΅ double prime πΆ double prime.
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We need to be careful that weβre using the second smaller triangle and asking how we go from this to the larger triangle.
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If we look at our list of possible transformations, the first three β translation, reflection, and rotation β keep the object and its image the same size.
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As the triangles here are different sizes, that means thereβs just one possible transformation, dilation.
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To describe a dilation, we need to find the center of dilation and the scale factor.
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We can find the scale factor relatively easily by looking at how the length on the image have increased from the length on the original shape.
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We can compare the lengths of π΄ double prime π΅ double prime and π΄ prime π΅ prime.
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We can see that on π΄ prime π΅ prime, the length is two units, and on the top length of π΄ double prime π΅ double prime, this is four units long.
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So, it looks like weβll have a scale factor of two.
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But itβs always worth checking some of the lengths on the other sides, just to be sure.
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The length π΅ prime πΆ prime is three units long and the length π΅ double prime πΆ double prime is six units long.
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And as thatβs twice as large, then weβve confirmed that the scale factor is two.
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Thereβs a nice, easy way to find the center of dilation.
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To do this, we create a ray between each vertex and its image.
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Here, we have a ray between π΅ prime and π΅ double prime and π΄ prime and π΄ double prime.
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We can do the same between πΆ prime and πΆ double prime.
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And the place where the rays converge will be the center of dilation, which once again will be the origin or the coordinate zero, zero.
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We put our answer into this statement form that this will be a dilation from the origin by a scale factor of two.
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Our final question asks if our triangles π΄π΅πΆ and π΄ double prime π΅ double prime πΆ double prime are similar.
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We can recall that similar means the same shape, but a different size.
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The angles remain the same, but the sides will be in proportion.
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So, in our first transformation from π΄π΅πΆ to π΄ prime π΅ prime πΆ prime, we didnβt change the size of these triangles, which means that they are congruent.
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And when we transformed π΄ prime π΅ prime πΆ prime to π΄ double prime π΅ double prime πΆ double prime, the image here did get larger.
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So, these two triangles would not be congruent.
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Each of the lengths in the image of π΄ double prime π΅ double prime πΆ double prime was in proportion to those in the triangles of π΄ prime π΅ prime πΆ prime.
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All corresponding pairs of angles are congruent.
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So, our triangles π΄π΅πΆ and π΄ double prime π΅ double prime πΆ double prime are similar.
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So, our answer for the final part of this question is: yes, these triangles are similar.