In the given figure, 𝐷𝐸 and 𝐵𝐶 are parallel. Use similarity to work out the value of 𝑥.
Because the line 𝐷𝐸 is parallel to the line 𝐵𝐶, these two triangles are similar. Triangle 𝐴𝐷𝐸 is a smaller version of triangle 𝐴𝐵𝐶. Because the angles are the same, they’re congruent, and so the sides of this triangle are in proportion. This means that there’s 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 𝐴 to 𝐵 is going to be the length of 𝐴𝐷 plus the length of 𝐷𝐵. That’s 𝑥 add three. And then the length from 𝐴 to 𝐶 is the length of 𝐴𝐸 plus the length from 𝐸 to 𝐶. That’s five add 𝑥 add two, and that’s 𝑥 add seven.
Because these triangles are similar, there’s a scale factor that we can find. This just means the ratio of the corresponding lines. For example, because the line 𝐴𝐸 corresponds with the line 𝐴𝐶, there’s a scale factor that we can multiply by the line 𝐴𝐸 to get the length of the line 𝐴𝐶. And then if we take the length of the line 𝐴𝐷, we should be able to multiply by that exact same scale factor to get the length of 𝐴𝐵. So this is the concept that we’re 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 𝐴𝐶 and 𝐴𝐸, which are corresponding sides, their scale factor is 𝑥 plus seven over five.
Now this should give us exactly the same value as the ratio of the sides 𝐴𝐵 and 𝐴𝐷. We can find the scale factor of the corresponding sides 𝐴𝐵 and 𝐴𝐷 again by doing the new length over the original length. That’s 𝑥 plus three over three. So what we’re saying is that these two scale factors should be exactly the same. So 𝑥 plus seven over five must be equal to 𝑥 plus three over three. We can then solve this by cross multiplying. We can then distribute the parentheses. That gives us three 𝑥 plus 21 equals five 𝑥 plus 15. Then subtracting three 𝑥 from both sides and then subtracting 15 from both sides gives us that six equals two 𝑥. And that gives us that 𝑥 is equal to three.
Note that, for this question, we could’ve chosen the new length to be from the smaller triangle and the original length to be from the bigger triangle. We would’ve just ended up with the numerator and the denominator the other way around in both the ratios. But we would’ve still ended up with exactly the same answer, 𝑥 equals three.