### Video Transcript

Find the length of line segment ๐ถ๐ต.

In this question, weโre given a diagram with two different triangles. Weโre given the lengths of two sides of each triangle. And we can also see from the markings that this angle, ๐ด๐ถ๐น, is the same size as this angle, ๐ท๐ต๐น. So, the line segment that weโre asked to work out is the one at the base of the diagram. Weโve got the length of part of it. We know that ๐ถ๐น is 21 units long and ๐น๐ต is two ๐ฅ plus eight. While we could just add these two lengths, for example, 21 plus two ๐ฅ plus eight would give us two ๐ฅ plus 29, but we can assume that what weโre really being asked here is for a numerical value for the length.

So, letโs have a look again at the diagram and see if thereโs any way we can work out this length, ๐น๐ต, perhaps by finding the value of ๐ฅ. A clue for the message comes from the fact that weโre given this pair of angles which are equal. We might then wonder if these triangles are perhaps similar or congruent. We remember that similar triangles have corresponding angles equal and corresponding sides in proportion. When weโre dealing with congruent triangles, congruent triangles have corresponding angles equal and corresponding sides equal.

However, going by the diagram, these two triangles arenโt the same size, so theyโre very likely not to be congruent. So, letโs check if theyโre similar. We can note firstly that our pairs of angles are equal. Angle ๐ด๐ถ๐น is equal to angle ๐ท๐ต๐น. Next, we could have a look at this angle ๐ด๐น๐ถ, which is marked as a right angle. Using the fact that the angles on a straight line sum to 180 degrees and the line ๐ต๐ถ is a straight line, this means that this angle of ๐ท๐น๐ต must also be 90 degrees, since 180 degrees subtract 90 also gives us 90 degrees. Therefore, we note that angle ๐ด๐น๐ถ is equal to angle ๐ท๐น๐ต.

This now means that weโve found two pairs of corresponding angles equal. This fulfills the AA or angle-angle similarity criterion. Now, weโve proven that triangle ๐ด๐น๐ถ is similar to triangle ๐ท๐น๐ต. So, letโs see how this helps us to work out the side length of ๐น๐ต. As these two triangles are similar, that means remember that corresponding sides are in proportion. If we look at the side ๐ด๐ถ, then the side which corresponds to it in triangle ๐ท๐น๐ต is this one, ๐ท๐ต. Then, another side on triangle ๐ด๐น๐ถ that we can look at is the length of ๐ถ๐น. The corresponding side on the other triangle is ๐น๐ต.

Because the triangles are similar, the sides are in proportion. And as that proportion is equal, then we can write that ๐ด๐ถ over ๐ท๐ต is equal to ๐ถ๐น over ๐น๐ต. We could also have written this statement with the fractions reversed. However, we need to make sure that we keep all the lengths of one triangle either as the numerators or the denominators and make sure we donโt mix them up.

What we do next is simply substitute in the length information that weโre given. This gives us 35 over seven ๐ฅ plus six equals 21 over two ๐ฅ plus eight. To solve this, we can begin by taking the cross product. So, we have 35 multiplied by two ๐ฅ plus eight equals 21 multiplied by seven ๐ฅ plus six. In the next step, we could go straight ahead and expand the parentheses on both sides of this equation. However, we might also notice that the values outside the parentheses are both multiples of seven. Dividing through by seven means that we can write it a little more simply. Five multiplied by two ๐ฅ plus eight equals three multiplied by seven ๐ฅ plus six.

We can now expand the parentheses giving us 10๐ฅ plus 40 equals 21๐ฅ plus 18. In order to keep a positive value of ๐ฅ, we can subtract 10๐ฅ from both sides. Then, we can subtract 18 from both sides, which leaves us with 22 equals 11๐ฅ. We can then divide through by 11 which gives us that two equals ๐ฅ and ๐ฅ is equal to two.

Itโs very tempting to stop here and think that weโve answered the question. But donโt forget we werenโt just asked for ๐ฅ. We were asked for the length of the line segment ๐ถ๐ต. Remember that we said that ๐ถ๐ต is equal to 21 plus this length of two ๐ฅ plus eight. We then need to substitute in the value that ๐ฅ is equal to two, giving us that ๐ถ๐ต is equal to 21 plus two times two plus eight simplifying to 33. Therefore, we can give the answer that line segment ๐ถ๐ต is equal to 33 length units.