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In this video, we will learn how to use the properties of similar polygons to find unknown angles, side lengths, scale factors, and perimeters.
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Firstly, we recall that a polygon is a closed shape with straight sides.
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So, for example, a triangle, a square, and a hexagon are all examples of polygons, whereas a circle isnโt as its sides are not all straight.
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Two polygons are said to be mathematically similar if one is an enlargement of the other.
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This means that two key things will be true.
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Firstly, corresponding angles in the two similar polygons are congruent or identical.
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Secondly, corresponding pairs of sides are in proportion.
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Another way of saying this is to say that corresponding pairs of sides are all in the same ratio.
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So in our example here, the length of ๐ด๐ต divided by the length of the corresponding side ๐ธ๐น gives the same result as if we divide ๐ต๐ถ by ๐น๐บ, and if we divide ๐ถ๐ท by ๐บ๐ป, and if we divide ๐ท๐ด by ๐ป๐ธ.
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If two polygons are similar, then we can express this using the notation on screen.
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Notice that the order of the letters is important.
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So if Iโve said ๐ด๐ต๐ถ๐ท is similar to ๐ธ๐น๐บ๐ป, then this means that the angle at ๐ด corresponds to the angle at ๐ธ.
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And the side ๐ถ๐ท corresponds to the side ๐บ๐ป.
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If two polygons are similar, then we can calculate the scale factor between them.
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This is the ratio of corresponding lengths and will always be a multiplier.
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Itโs the value we need to multiply the length in one polygon by to find the corresponding length in the other polygon.
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We can calculate the scale factor in either direction.
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For example, if we need to multiply the lengths of ๐ด๐ต๐ถ๐ท by ๐ in order to give the corresponding lengths in ๐ธ๐น๐บ๐ป, then in order to go the other way, weโd need to multiply the lengths of ๐ธ๐น๐บ๐ป by one over ๐ to give the corresponding lengths in ๐ด๐ต๐ถ๐ท.
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Itโs important to notice that the scale factor is always a multiplier.
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So, in formulae, we may think of going from the larger polygon to the smaller polygon as dividing by ๐.
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But our scale factor will be the multiplier one over ๐.
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Letโs now look at a question where we need to calculate the scale factor between two similar polygons and then determine the length of a side.
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Given that the rectangles shown are similar, what is ๐ฅ?
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Weโre told in the question that these two rectangles are mathematically similar, which means that two things are true.
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Firstly, all pairs of corresponding angles between the two rectangles are congruent.
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Now, for a pair of rectangles, this is true even for nonsimilar rectangles as all the interior angles in a rectangle are 90 degrees.
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Secondly, and more importantly here, all pairs of corresponding sides are in proportion.
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Weโre looking to find the value of ๐ฅ which represents a side length in the larger rectangle.
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We therefore need to know the scale factor ๐ that is the multiplier that takes us from the smaller rectangle to the larger.
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We can use any pair of corresponding sides to work out the scale factor.
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Itโs equal to the new length divided by the original length.
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From the figure, we can see that we have a pair of corresponding sides of 29 centimeters on the smaller rectangle and 58 centimeters on the larger.
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We therefore divide the new length of 58 by the original length of 29, giving a scale factor of 58 over 29, which simplifies to two.
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Remember, the scale factor is always a multiplier.
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So this tells us that the lengths on the larger rectangle are all twice the corresponding lengths on the smaller rectangle.
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To work out the value of ๐ฅ then, we need to take the corresponding length on the smaller rectangle, 26, and multiply it by the scale factor of two, which gives 52.
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So we found the value of ๐ฅ.
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If we had wanted to calculate a length on the smaller rectangle rather than one on the larger rectangle, we couldโve used the scale factor of one-half.
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Thatโs the reciprocal of two.
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This couldโve been found by dividing the length of 29 centimeters by the corresponding length of 58 centimeters.
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We can check our value for ๐ฅ by multiplying it by one-half, so 52 multiplied by one-half or half of 52, which is indeed equal to 26.
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In formulae, we may think of this as dividing by two.
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But remember, the scale factor is always a multiplier, so weโre using a multiplier of one-half.
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Letโs consider another example.
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If the two following polygons are similar, find the value of ๐ฅ.
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Weโre told that these two polygons are mathematically similar, which means that two key things are true.
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Firstly, corresponding pairs of angles are congruent.
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And secondly, corresponding pairs of sides are in proportion or in the same ratio.
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We want to calculate this ratio or scale factor.
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And in this case, as ๐ฅ represents the length on the smaller polygon, weโre working in this direction.
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The scale factor can be calculated from any corresponding pair of sides by dividing the new length by the original length.
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We have a corresponding pair of side lengths of 34 centimeters and 85 centimeters.
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So dividing the new length, thatโs the length which is in the same polygon as the one we want to calculate, by the original length, we have a scale factor of 34 over 85.
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This can be simplified by dividing both the numerator and denominator by 17 to give two-fifths.
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Notice that this value feels right.
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Weโre moving from the larger polygon to the smaller one.
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So the length should be getting smaller.
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Our scale factor is a fractional value less than one.
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And when we multiply something by a fractional value less than one, such as two-fifths, it will become smaller.
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To calculate the value of ๐ฅ then, we need to take the corresponding length on the larger polygon, which is 75 centimeters, and multiply it by our scale factor of two-fifths.
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We can simplify this calculation by canceling a factor of five from both the numerator and denominator to give ๐ฅ equals 15 multiplied by two, which is of course 30.
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So weโve found the value of ๐ฅ.
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We can check our answer by confirming that the ratio of corresponding pair of sides is indeed the same.
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We have 30 over 75 and 34 over 85.
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Both fractions do indeed cancel to our scale factor of two-fifths.
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And so this confirms that our answer is correct.
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In our next example, weโll see how we can test whether or not two polygons are similar.
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Is polygon ๐ด๐ต๐ถ๐ท similar to polygon ๐บ๐น๐ธ๐?
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From the figure, we can see that the two polygons weโve been given are each parallelograms.
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So we can conclude that they are at least the same type of shape to begin with.
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To determine whether theyโre similar, we need to test two things.
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Firstly, we need to test whether corresponding pairs of angles are congruent.
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And secondly, we need to test whether corresponding pairs of sides are in proportion or in the same ratio.
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Now, itโs important to remember that when weโre working with similar polygons, the order of the letters is important.
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So if these polygons are similar, then the angle at ๐ด will correspond to the angle at ๐บ.
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The angle at ๐ต will correspond to the angle at ๐น, and so on.
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We can therefore deduce that the polygons have been drawn in the same orientation.
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Letโs consider the angles first of all then.
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In polygon ๐บ๐น๐ธ๐, weโve been given a marked angle of 110 degrees.
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And in polygon ๐ด๐ต๐ถ๐ท, weโve been given a marked angle of 70 degrees.
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One thing we do know about parallelograms is that their opposite angles are equal.
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So in parallelogram ๐บ๐น๐ธ๐, the angle at ๐ will be 110 degrees.
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And in parallelogram ๐ด๐ต๐ถ๐ท, the angle at ๐ถ will be 70 degrees.
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Letโs consider the angle at ๐บ in the polygon ๐บ๐น๐ธ๐.
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By extending the line ๐น๐บ, we now see that we have two parallel lines ๐๐บ and ๐ธ๐น and a transversal ๐บ๐น.
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We know that corresponding angles in parallel lines are equal, which means that the angle above the line ๐๐บ will be equal to the angle above the line ๐ธ๐น.
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Itโs 110 degrees.
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We also know that angles on a straight line sum to 180 degrees, which means the angle below ๐๐บ will be 180 minus 110.
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Itโs 70 degrees.
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This shows us that the angle at ๐บ in polygon ๐บ๐น๐ธ๐ is equal to the angle at ๐ด in polygon ๐ด๐ต๐ถ๐ท.
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We already said that opposite angles in parallelograms are equal.
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So the angle at ๐ธ is also 70 degrees, which is equal to the angle at ๐ถ in polygon ๐ด๐ต๐ถ๐ท.
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We could use the same logic in parallelogram ๐ด๐ต๐ถ๐ท to show that the angles at ๐ต and ๐ท are each 110 degrees, which are the same as the angles at ๐น and ๐ in the larger polygon.
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Weโve shown then that all pairs of corresponding angles are indeed congruent.
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So our answer to the first check is yes.
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Letโs now consider whether corresponding pairs of sides are in proportion.
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Firstly, from the figure, we can see weโve been given side length ๐ด๐ต; itโs 13 centimeters.
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And if the two polygons are similar, this will correspond to ๐บ๐น.
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We havenโt been given the length of ๐บ๐น.
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But we know that opposite sides in a parallelogram are equal in length.
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So it will be the same as ๐๐ธ.
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Comparing the ratio of these two sides then, we find that ๐ด๐ต over ๐บ๐น is equal to 13 over 26, which simplifies to one-half.
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The other pair of potentially corresponding sides weโve been given are ๐ต๐ถ and ๐ธ๐น.
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The ratio here is 11.5 over 23, which again simplifies to one-half.
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Now, Iโve written ๐ธ๐น here.
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But really, if weโre to be consistent with the order of letters, then we should really write ๐น๐ธ as point ๐น corresponds to point ๐ต and point ๐ธ corresponds to point ๐ถ.
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However, in calculating the ratio, the length ๐ธ๐น is of course the same as the length ๐น๐ธ.
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So it makes no practical difference.
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The ratio of these pairs of corresponding sides is therefore the same.
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As opposite sides in a parallelogram are equal in length, the same will be true for the remaining two pairs of corresponding sides.
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And so the answer to our second check, โare corresponding pairs of sides in proportion?โ, is also yes.
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Hence, both of the criteria for these two polygons to be similar are fulfilled.
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And so we can answer, yes, polygon ๐ด๐ต๐ถ๐ท is similar to polygon ๐บ๐น๐ธ๐.
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Letโs now consider another example in which we prove similarity of two polygons.
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Are these two polygons similar?
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If yes, find the scale factor from ๐๐๐๐ฟ to ๐ด๐ต๐ถ๐ท.
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To determine whether these two polygons are similar, we need to consider two questions.
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Firstly, are corresponding pairs of angles congruent?
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And secondly, are corresponding pairs of sides in proportion?
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Remember that the order of the letters is important.
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So if these two polygons are similar, then the angle at ๐ will correspond with the angle at ๐ด.
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And, for example, the side ๐๐ฟ will correspond with the side ๐ถ๐ท.
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Letโs consider the angles first.
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And looking at the figure, we can see that three of the angles in each polygon have been marked using different notation.
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The single line marking the angle at ๐ฟ and the angle at ๐ท indicates that these two angles are equal.
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The pair of lines at angle ๐ and angle ๐ด indicates that these two angles are equal.
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In the same way, the triple line at angle ๐ and angle ๐ถ indicates that these two angles are equal.
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But what about the final angle in each polygon?
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Well, each of these polygons have four sides.
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They are quadrilaterals.
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And we know that the sum of the interior angles in any quadrilateral is 360 degrees.
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If all of the other three angles in the two quadrilaterals are the same, then in each case, we will have the same amount left over from the total of 360 degrees to form the final angle.
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And therefore, we can also say that angle ๐ is equal to angle ๐ต.
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So weโve seen that all pairs of corresponding angles are indeed congruent between the two polygons.
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So the answer to our first check is yes.
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Next, letโs consider the sides.
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And we need to check whether corresponding pairs of sides are in proportion.
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That is, do we get the same value when we divide each side of one polygon by the corresponding side of the other?
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Weโve been given all of the side lengths we need.
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So we can substitute them and work out what each of these ratios simplify to.
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In fact, they are all equal to four-fifths or 0.8.
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And so corresponding pairs of sides are indeed in proportion.
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As weโve answered yes to both statements, we can conclude that the two polygons are indeed similar.
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Weโre now asked to find the scale factor from ๐๐๐๐ฟ to ๐ด๐ต๐ถ๐ท.
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Thatโs traveling in this direction.
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Remember, the scale factor is the value we multiply the lengths on the first polygon by in order to give the corresponding lengths on the second.
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And in fact, weโve already worked this out.
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Itโs the value we get when we divide a new length, thatโs a length on the polygon weโre going to, by an original length.
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Thatโs the corresponding length on the polygon weโre coming from.
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Weโve already seen that when we divide each length on ๐ด๐ต๐ถ๐ท by the corresponding length on ๐๐๐๐ฟ, we get the value four-fifths.
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So this is our scale factor.
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Notice that this value does make logical sense.
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The lengths on ๐ด๐ต๐ถ๐ท are smaller than the lengths on ๐๐๐๐ฟ.
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So our scale factor should be a fractional value less than one.
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So weโve completed the problem.
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We found that these two polygons are similar.
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And the scale factor going from ๐๐๐๐ฟ to ๐ด๐ต๐ถ๐ท, thatโs from the larger polygon to the smaller, as a decimal is 0.8.
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Just as a reminder, if we were going back the other way, so finding the scale factor from ๐ด๐ต๐ถ๐ท to ๐๐๐๐ฟ, then we would have the reciprocal.
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The reciprocal of a fraction can be found by flipping it, so swapping its numerator and denominator.
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The scale factor going in the other direction would be the multiplier of five over four.
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Again, this makes sense.
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Five over four is a little bit bigger than one.
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Itโs 1.25 as a decimal.
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And so weโre multiplying the lengths on ๐ด๐ต๐ถ๐ท by something bigger than one, which will give larger values for the corresponding lengths on ๐๐๐๐ฟ.
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Letโs now summarize some of the key points from this video.
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Firstly, two polygons are similar if corresponding pairs of angles are congruent and all corresponding pairs of sides are in proportion.
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To calculate the scale factor between two similar polygons, we can use the lengths of any pair of corresponding sides.
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And we divide the new length, thatโs the length on the polygon weโre going to, by the corresponding length on the original.
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Thatโs the polygon weโre coming from.
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Remember that the scale factor is always a multiplier.
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When weโre going from the smaller shape to the larger shape, we will always have a scale factor greater than one.
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When going in the other direction, so going from the larger shape to the smaller shape, the scale factor will be the reciprocal of the scale factor working in the other direction.
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It will be one over ๐.
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And in this case, the scale factor will be less than one.
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In formulae, we may think of this as dividing by the previous scale factor of ๐.
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But our scale factor should really be a multiplier.
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Once we know that two polygons are similar, we can calculate the measures of any missing angles.
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And we can use scale factors to calculate the lengths of any missing sides, provided weโve been given the corresponding information on the other polygon.