Question Video: Finding the Length of a Side in a Polygon given the Corresponding Side’s Length in a Similar Polygon and the Ratio between Their Areas | Nagwa Question Video: Finding the Length of a Side in a Polygon given the Corresponding Side’s Length in a Similar Polygon and the Ratio between Their Areas | Nagwa

Question Video: Finding the Length of a Side in a Polygon given the Corresponding Side’s Length in a Similar Polygon and the Ratio between Their Areas Mathematics • First Year of Secondary School

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The two shown figures are similar. Given that the area of the blue shape is 19.32 cm² and that of the yellow one is 77.28 cm², find the value of 𝑥.

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Video Transcript

The two shown figures are similar. Given that the area of the blue shape is 19.32 square centimeters and that of the yellow one is 77.28 square centimeters, find the value of 𝑥.

The figure shows us two kites. And we’re given the area that the blue shape — that’s the smaller one — is 19.32 square centimeters. The area of the yellow one is 77.28 square centimeters. We’re asked to find the value of 𝑥, which is one of the side lengths on the larger yellow figure. We can do this by using the information that these two kites are similar.

Similar shapes have the same number of sides, their corresponding angles are congruent, and their corresponding sides are in proportion. We can use the given information about the area of each shape and use this to work out the area ratio. Once we know this, we can find their length ratio and work out the value of 𝑥. We will use the fact that if the length ratio of two similar shapes is 𝑎 to 𝑏, then the ratio of their areas is 𝑎 squared to 𝑏 squared. Since the length of 5.3 and 𝑥 are of corresponding sides, then we can write that the length ratio of the blue shape to the yellow shape is 5.3 to 𝑥. Using the statement above, we could then say that the area ratio of the blue to the yellow shape must be 5.3 squared to 𝑥 squared.

Using the given information that the two areas here are 19.32 and 77.28, we also know that the areas are in the ratio of 19.32 to 77.28. We know that the areas must be in the same ratio as the area ratio given above. We can then use this pair of equivalent ratios to work out the value of 𝑥 squared and hence the value of 𝑥. We can write this fractionally to help us solve it, since 5.3 squared over 19.32 must be equal to the proportion of 𝑥 squared over 77.28. We can even work out the value of 5.3 squared first; it’s 28.09. By cross multiplying, we have 19.32𝑥 squared is equal to 28.09 times 77.28. We can then divide both sides by 19.32. We can then use our calculators to evaluate 28.09 times 77.28 over 19.32, giving us 𝑥 squared is equal to 112.36.

Remember that we now need to work out 𝑥. And so, we take the square root of both sides of the equation. And since 𝑥 is a length, we know that we’re only interested in the positive value of the square root of 112.36, which is 10.6. We don’t need to include the units of centimeters since the length was defined as 𝑥 centimeters. And so, by using the relationship between the area and length ratios of two similar shapes, we have calculated that 𝑥 is equal to 10.6.

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