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