ShmoopTube

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Similar Figures, a la Shmoop.

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We here at Shmoop have been making great strides on our new shrink ray design. [Girls working at shmoop]

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We just have to…work out a few final kinks.

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But it sure has taught us a lot about similar figures…

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…which are figures that have the same shape

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as one another, but not necessarily the same size. [Two different size cubes spin]

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Let’s take the Bermuda Triangle, for example. It’s rumored that any ship that sails into [Ship sailing into the bermuda triangle]

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this part of the ocean…instantly sinks.

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Now let’s say the Bermuda Triangle is twenty miles by twelve miles by eight miles.

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And in order for his cousin, “Little” Bermuda Triangle, to be a similar figure…

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…she would have to have the same exact proportions. [Two different size triangles]

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If each side of the little Bermuda Triangle is exactly proportional to the corresponding

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side of the real Bermuda triangle… …the two triangles would be similar figures.

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So…if Little Bermuda is 10 by 6 by 4…the triangles would be similar…

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…since each distance is exactly half that of the original.

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If Little Bermuda is ten by six by two… …then our hopes are sunk. [Hope sinks into ocean]

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Because while two of the three sides are half the length of the original…

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…the final side is only a quarter of the distance of its big cousin.

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Another way to determine whether or not figures are similar is to check their angles. [Shape with legs dancing]

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Ahem, said angles people, listen up.

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If all of the corresponding angles in two triangles are the same, then the figures are similar.

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Like…if this 35 degree angle matches this 35 degree angle… [Two 35 degree angle triangles]

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…and this 45 degree angle matches this 45 degree angle…

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…and this 100 degree angle matches this 100 degree angle.

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Boom. Similar.

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So we know that similar figures have sides that are proportionate and angles that are congruent.

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Why might you need to pop this information into your brain? [Zombie man appears]

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Say someday you become an architect, and you’d [Architect making measurements]

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like to build a model of a proposed structure that isn’t, um…life-size.

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That would be an awful waste of Styrofoam.

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Now let’s say you want to know how tall an existing building is. [A tall skyscraper]

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Maybe you need this information to rappel down the side of some building…

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we won’t get too nosy.

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Well, you can measure your own shadow. You know how tall you are, right? [Man stood measuring his shadow]

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Now you can measure the building’s shadow. The proportion of actual height to the length [Man stood beside skyscraper]

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of its shadow will be the same for you as for the building.

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So…if you cast a three foot long shadow, and you’re six feet tall… looks like the

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ratio is 2 to 1. So if the building casts a twenty foot long

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shadow…the building must be forty feet high. Same ratio – 2 to 1.

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If we look at these as triangles, we can see that we’re dealing with right angles…

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…and as object and shadow height increase, so does the imaginary hypotenuse. [Man and skyscraper with 90 degree triangles in front of them]

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It’s really just two similar triangles…and we can be sure that our determination of the

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building’s height is accurate.

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You may now rappel down it with complete confidence. Good luck avoiding the searchlights though… [Man rappelling down a skyscraper]

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