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Given that 𝑄𝑅𝑃𝑀 is similar to 𝑆𝑉𝑍𝑊, find the value of 𝑥.
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Note that the approximation symbol here means that the two rectangles are similar.
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When two polygons are similar, we know that their corresponding sides are proportional.
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In this question, we have corresponding sides 𝑄𝑀 and 𝑆𝑊 along with 𝑀𝑃 and 𝑊𝑍.
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This means that the ratio of six 𝑥 minus 16 to 13 will be the same as nine 𝑥 minus 33 to 15.
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Writing this in fractional form gives us six 𝑥 minus 16 over nine 𝑥 minus 33 is equal to 13 over 15.
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In order to calculate 𝑥, we could cross multiply immediately.
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However, it is often useful to try and simplify the fractions first.
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The numerator and denominator on the left-hand side can be factored.
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Six 𝑥 minus 16 is equal to two multiplied by three 𝑥 minus eight.
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And nine 𝑥 minus 33 is equal to three multiplied by three 𝑥 minus 11.
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The denominators have a common factor of three, so we can divide both of these by three.
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Cross multiplying at this stage gives us 10 multiplied by three 𝑥 minus eight is equal to 13 multiplied by three 𝑥 minus 11.
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We get the 10 by multiplying five by two.
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Redistributing our parentheses gives us 30𝑥 minus 80 is equal to 39𝑥 minus 143.
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Adding 143 to both sides of this equation gives us 30𝑥 plus 63 is equal to 39𝑥.
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Subtracting 30𝑥 from both sides gives us 63 is equal to nine 𝑥.
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Finally, dividing both sides of this equation by nine gives us a value of 𝑥 equal to seven.
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We can then check this answer by substituting 𝑥 equals seven into our expressions on the first rectangle.
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Six multiplied by seven is equal to 42, and subtracting 16 gives us 26.
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Nine multiplied by seven is equal to 63.
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Subtracting 33 from this gives us 30.
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We can therefore see that our first rectangle is twice the size of our second rectangle as 26 is double 13 and 30 is double 15.
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The scale factor to get from rectangle 𝑄𝑅𝑃𝑀 to 𝑆𝑉𝑍𝑊 is a half as the corresponding sides of the second rectangle are half the size of the first one.