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 • 11th Grade

In the figure, ๐ด๐ท = 10, ๐ด๐ธ = 5, ๐ต๐ท = 4๐‘ฅ + 2, and ๐ธ๐ถ = ๐‘ฅ + 3. What are the lengths of the line segment ๐ด๐ต and the line segment ๐ธ๐ถ?

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Video Transcript

In the figure, ๐ด๐ท equals 10, ๐ด๐ธ equals five, ๐ต๐ท equals four ๐‘ฅ plus two, and ๐ธ๐ถ equals ๐‘ฅ plus three. What are the lengths of the line segment ๐ด๐ต and the line segment ๐ธ๐ถ?

Well, the first thing we want to do is mark on all the information weโ€™ve been given. So weโ€™ve got ๐ด๐ท equals 10, ๐ด๐ธ equals five, ๐ต๐ท equals four ๐‘ฅ plus two, and ๐ธ๐ถ equals ๐‘ฅ plus three. Now, if we take a look at the shape itself, then what we actually have are two triangles, triangle ๐ด๐ท๐ธ and triangle ๐ด๐ต๐ถ. 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 ๐ด๐ถ, we can say that the angles at ๐ถ and ๐ธ are corresponding. Similarly, if we look at the line ๐ด๐ต, which transverses our two parallel lines, then this means that weโ€™re gonna have corresponding angles at ๐ท and ๐ต, so these are gonna be the same. And then finally, both triangles have a shared angle at ๐ด.

Okay, so what does this mean? Well, what it tells us is that using the proof AAA or angle-angle-angle, then triangle ๐ด๐ท๐ธ is similar to triangle ๐ด๐ต๐ถ. 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 ๐ด๐ต over ๐ด๐ท is gonna be equal to ๐ด๐ถ over ๐ด๐ธ. And as we said, itโ€™s because our corresponding sides are proportional. So therefore, what weโ€™re gonna have is ๐ด๐ต, which is gonna be four ๐‘ฅ plus two plus 10 because itโ€™s ๐ท๐ต and ๐ด๐ท, and then over 10, because thatโ€™s our ๐ด๐ท. And itโ€™s gonna be equal to ๐ด๐ถ, which is ๐‘ฅ plus three plus five, over ๐ด๐ธ, which is five.

Okay, great. So now what we could do is solve this equation to find out what ๐‘ฅ is. So next, what weโ€™re gonna do is multiply through both sides by 10. And thatโ€™s because we want to remove the fractional element of our equation. And when we do that, weโ€™re gonna get four ๐‘ฅ plus 12 equals two multiplied by ๐‘ฅ plus eight. And then if you divide three by two, you get two ๐‘ฅ plus six equals ๐‘ฅ plus eight. So then, if we subtract ๐‘ฅ and subtract six from each side of the equation, weโ€™re gonna get ๐‘ฅ equals two.

Okay, great. So weโ€™ve now found our value for ๐‘ฅ. Well, if we take a look at ๐‘ฅ, what we can see is that we want to find the lengths of the line segment ๐ด๐ต and the line segment ๐ธ๐ถ. Well, weโ€™ve already identified that ๐ด๐ต is equal to four ๐‘ฅ plus two plus 10, which is four ๐‘ฅ plus 12. Well, as weโ€™ve identified that ๐‘ฅ is equal to two, weโ€™re gonna substitute this in. So weโ€™re gonna get four multiplied by two plus 12. So weโ€™ve answered the first part of the question because ๐ด๐ต is equal to 20.

Okay, so now letโ€™s move on to ๐ธ๐ถ. Well, we can see that ๐ธ๐ถ is ๐‘ฅ plus three. So if we substitute in ๐‘ฅ equals two, itโ€™s gonna be two plus three, which is gonna give us five. So therefore, we can say that the line segments ๐ด๐ต and ๐ธ๐ถ have lengths of 20 and five, respectively.

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