### Video Transcript

In the given figure, ๐ด๐ต equals 35, ๐ด๐ถ equals 30, and ๐ถ๐ท equals 12. If ๐ต๐ท equals ๐ฅ plus 10, what is the value of ๐ฅ?

Weโve been given a diagram of a triangle and the lengths of various lines within this triangle. Letโs first add this information to the diagram. The question asked us to find the value of ๐ฅ, which forms part of the expression for ๐ต๐ท. Letโs think about how to approach this problem.

The line ๐ด๐ท is a bisector of the angle ๐ถ๐ด๐ต. We can see this because the two parts of the angle have each been marked with a single blue arc, indicating that they are equal. Therefore, we need to approach this problem using facts about angle bisectors. The angle bisector divides the opposite side of the triangle ๐ถ๐ต into two parts, ๐ถ๐ท and ๐ท๐ต.

The ratio between the lengths of these two parts is the same as the ratio of the lengths of the other two sides of the triangle. Or, in other words, for this triangle, the ratio we get when we divide by ๐ต๐ท by ๐ถ๐ท is the same as the ratio we get when we divide ๐ด๐ต by ๐ด๐ถ. In each case, this is the pink side divided by the green side.

We can substitute in the values or, in the case of ๐ต๐ท, the expression for each of these sides to give an equation that we can solve to find the value of ๐ฅ. ๐ต๐ท over ๐ถ๐ท becomes ๐ฅ plus 10 over 12. ๐ด๐ต over ๐ด๐ถ becomes 35 over 30. This fraction can be simplified by dividing both the numerator and denominator by five to give a simplified fraction of seven over six.

Now letโs think about how to solve this equation. We have a 12 in the denominator of one fraction and a six in the denominator of the other. Multiplying both sides of the equation by 12 will eliminate both these denominators. The 12 that now appears in the numerator on the right-hand side will cancel with the six in the denominator to give an overall factor of two.

So weโre left with ๐ฅ plus 10 is equal to seven multiplied by two, which is 14. The final step in solving this equation is we need to subtract 10 from both sides. This gives ๐ฅ is equal to four. So we found the value of ๐ฅ. Remember, the key fact that we used in this question is that an angle bisector divides the opposite side of a triangle in the same ratio as the ratio that exists between the other two sides of the triangle.