Question Video: Finding the Side Length of a Quadrilateral given the Corresponding Side’s Length in a Similar Quadrilateral Mathematics

Given that 𝑄𝑅𝑃𝑀 ∼ π‘†π‘‰π‘π‘Š, find the value of π‘₯.

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Video Transcript

Given that 𝑄𝑅𝑃𝑀 is similar to π‘†π‘‰π‘π‘Š, find the value of π‘₯.

Note that the approximation symbol here means that the two rectangles are similar. When two polygons are similar, we know that their corresponding sides are proportional. In this question, we have corresponding sides 𝑄𝑀 and π‘†π‘Š along with 𝑀𝑃 and π‘Šπ‘. This means that the ratio of six π‘₯ minus 16 to 13 will be the same as nine π‘₯ minus 33 to 15. Writing this in fractional form gives us six π‘₯ minus 16 over nine π‘₯ minus 33 is equal to 13 over 15.

In order to calculate π‘₯, we could cross multiply immediately. However, it is often useful to try and simplify the fractions first. The numerator and denominator on the left-hand side can be factored. Six π‘₯ minus 16 is equal to two multiplied by three π‘₯ minus eight. And nine π‘₯ minus 33 is equal to three multiplied by three π‘₯ minus 11. The denominators have a common factor of three, so we can divide both of these by three.

Cross multiplying at this stage gives us 10 multiplied by three π‘₯ minus eight is equal to 13 multiplied by three π‘₯ minus 11. We get the 10 by multiplying five by two. Redistributing our parentheses gives us 30π‘₯ minus 80 is equal to 39π‘₯ minus 143. Adding 143 to both sides of this equation gives us 30π‘₯ plus 63 is equal to 39π‘₯. Subtracting 30π‘₯ from both sides gives us 63 is equal to nine π‘₯. Finally, dividing both sides of this equation by nine gives us a value of π‘₯ equal to seven.

We can then check this answer by substituting π‘₯ equals seven into our expressions on the first rectangle. Six multiplied by seven is equal to 42, and subtracting 16 gives us 26. Nine multiplied by seven is equal to 63. Subtracting 33 from this gives us 30. 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. The scale factor to get from rectangle 𝑄𝑅𝑃𝑀 to π‘†π‘‰π‘π‘Š is a half as the corresponding sides of the second rectangle are half the size of the first one.

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