WEBVTT
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In the given figure, ๐ท๐ธ and ๐ต๐ถ are parallel.
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Use similarity to work out the value of ๐ฅ.
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Because the line ๐ท๐ธ is parallel to the line ๐ต๐ถ, these two triangles are similar.
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Triangle ๐ด๐ท๐ธ is a smaller version of triangle ๐ด๐ต๐ถ.
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Because the angles are the same, theyโre congruent, and so the sides of this triangle are in proportion.
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This means that thereโs a value, a scale factor, which we can multiply each length of the smaller triangle by to get the corresponding length of the bigger triangle.
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We can draw these triangles side by side if it helps.
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The length from ๐ด to ๐ต is going to be the length of ๐ด๐ท plus the length of ๐ท๐ต.
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Thatโs ๐ฅ add three.
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And then the length from ๐ด to ๐ถ is the length of ๐ด๐ธ plus the length from ๐ธ to ๐ถ.
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Thatโs five add ๐ฅ add two, and thatโs ๐ฅ add seven.
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Because these triangles are similar, thereโs a scale factor that we can find.
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This just means the ratio of the corresponding lines.
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For example, because the line ๐ด๐ธ corresponds with the line ๐ด๐ถ, thereโs a scale factor that we can multiply by the line ๐ด๐ธ to get the length of the line ๐ด๐ถ.
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And then if we take the length of the line ๐ด๐ท, we should be able to multiply by that exact same scale factor to get the length of ๐ด๐ต.
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So this is the concept that weโre going to use to answer this problem.
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The ratio of corresponding lengths can be found by taking the new length and dividing it by the original length.
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So thinking about the lengths ๐ด๐ถ and ๐ด๐ธ, which are corresponding sides, their scale factor is ๐ฅ plus seven over five.
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Now this should give us exactly the same value as the ratio of the sides ๐ด๐ต and ๐ด๐ท.
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We can find the scale factor of the corresponding sides ๐ด๐ต and ๐ด๐ท again by doing the new length over the original length.
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Thatโs ๐ฅ plus three over three.
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So what weโre saying is that these two scale factors should be exactly the same.
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So ๐ฅ plus seven over five must be equal to ๐ฅ plus three over three.
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We can then solve this by cross multiplying.
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We can then distribute the parentheses.
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That gives us three ๐ฅ plus 21 equals five ๐ฅ plus 15.
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Then subtracting three ๐ฅ from both sides and then subtracting 15 from both sides gives us that six equals two ๐ฅ.
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And that gives us that ๐ฅ is equal to three.
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Note that, for this question, we couldโve chosen the new length to be from the smaller triangle and the original length to be from the bigger triangle.
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We wouldโve just ended up with the numerator and the denominator the other way around in both the ratios.
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But we wouldโve still ended up with exactly the same answer, ๐ฅ equals three.