Question Video: Finding the Length of a Side in a Triangle given the Corresponding Side in a Similar Triangle and the Similarity Ratio between Them Mathematics • First Year of Secondary School

Video Transcript

In the figure, 𝐴𝐷 equals 10, 𝐴𝐸 equals five, 𝐵𝐷 equals four 𝑥 plus two, and 𝐸𝐶 equals 𝑥 plus three. What are the lengths of the line segment 𝐴𝐵 and the line segment 𝐸𝐶?

Well, the first thing we want to do is mark on all the information we’ve been given. So we’ve got 𝐴𝐷 equals 10, 𝐴𝐸 equals five, 𝐵𝐷 equals four 𝑥 plus two, and 𝐸𝐶 equals 𝑥 plus three. Now, if we take a look at the shape itself, then what we actually have are two triangles, triangle 𝐴𝐷𝐸 and triangle 𝐴𝐵𝐶. And we also have two parallel lines. But we can use something about these parallel lines to tell us something about our triangles.

Well, as our parallel lines are transversed by the line 𝐴𝐶, we can say that the angles at 𝐶 and 𝐸 are corresponding. Similarly, if we look at the line 𝐴𝐵, which transverses our two parallel lines, then this means that we’re gonna have corresponding angles at 𝐷 and 𝐵, so these are gonna be the same. And then finally, both triangles have a shared angle at 𝐴.

Okay, so what does this mean? Well, what it tells us is that using the proof AAA or angle-angle-angle, then triangle 𝐴𝐷𝐸 is similar to triangle 𝐴𝐵𝐶. Well, what do we know about similar triangles? Well, actually, what it means is that they are in fact an enlargement of one another. So therefore, corresponding sides are always proportional.

So therefore, if we take a look at our triangle, we can say that 𝐴𝐵 over 𝐴𝐷 is gonna be equal to 𝐴𝐶 over 𝐴𝐸. And as we said, it’s because our corresponding sides are proportional. So therefore, what we’re gonna have is 𝐴𝐵, which is gonna be four 𝑥 plus two plus 10 because it’s 𝐷𝐵 and 𝐴𝐷, and then over 10, because that’s our 𝐴𝐷. And it’s gonna be equal to 𝐴𝐶, which is 𝑥 plus three plus five, over 𝐴𝐸, which is five.

Okay, great. So now what we could do is solve this equation to find out what 𝑥 is. So next, what we’re gonna do is multiply through both sides by 10. And that’s because we want to remove the fractional element of our equation. And when we do that, we’re gonna get four 𝑥 plus 12 equals two multiplied by 𝑥 plus eight. And then if you divide three by two, you get two 𝑥 plus six equals 𝑥 plus eight. So then, if we subtract 𝑥 and subtract six from each side of the equation, we’re gonna get 𝑥 equals two.

Okay, great. So we’ve now found our value for 𝑥. Well, if we take a look at 𝑥, what we can see is that we want to find the lengths of the line segment 𝐴𝐵 and the line segment 𝐸𝐶. Well, we’ve already identified that 𝐴𝐵 is equal to four 𝑥 plus two plus 10, which is four 𝑥 plus 12. Well, as we’ve identified that 𝑥 is equal to two, we’re gonna substitute this in. So we’re gonna get four multiplied by two plus 12. So we’ve answered the first part of the question because 𝐴𝐵 is equal to 20.

Okay, so now let’s move on to 𝐸𝐶. Well, we can see that 𝐸𝐶 is 𝑥 plus three. So if we substitute in 𝑥 equals two, it’s gonna be two plus three, which is gonna give us five. So therefore, we can say that the line segments 𝐴𝐵 and 𝐸𝐶 have lengths of 20 and five, respectively.