Question Video: Finding the Length of a Rectangle’s Side given Its Corresponding One in a Similar Rectangle Mathematics • 8th Grade

Given that the rectangles shown are similar, what is 𝑥?

02:23

Video Transcript

Given that the rectangles shown are similar, what is 𝑥?

We’re told in the question that these two rectangles are mathematically similar, which means that two things are true. Firstly, all pairs of corresponding angles between the two rectangles are congruent. Now, for a pair of rectangles, this is true even for nonsimilar rectangles as all the interior angles in a rectangle are 90 degrees. Secondly, and more importantly here, all pairs of corresponding sides are in proportion.

We’re looking to find the value of 𝑥 which represents a side length in the larger rectangle. We therefore need to know the scale factor 𝑆 that is the multiplier that takes us from the smaller rectangle to the larger. We can use any pair of corresponding sides to work out the scale factor. It’s equal to the new length divided by the original length. 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. 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.

Remember, the scale factor is always a multiplier. So this tells us that the lengths on the larger rectangle are all twice the corresponding lengths on the smaller rectangle. To work out the value of 𝑥 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. So we found the value of 𝑥. If we had wanted to calculate a length on the smaller rectangle rather than one on the larger rectangle, we could’ve used the scale factor of one-half. That’s the reciprocal of two. This could’ve been found by dividing the length of 29 centimeters by the corresponding length of 58 centimeters.

We can check our value for 𝑥 by multiplying it by one-half, so 52 multiplied by one-half or half of 52, which is indeed equal to 26. In formulae, we may think of this as dividing by two. But remember, the scale factor is always a multiplier, so we’re using a multiplier of one-half.

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