In the given figure, 𝐴𝐵 equals 35, 𝐴𝐶 equals 30, and 𝐶𝐷 equals 12. If 𝐵𝐷 equals 𝑥 plus 10, what is the value of 𝑥?
We’ve been given a diagram of a triangle and the lengths of various lines within this triangle. Let’s first add this information to the diagram. The question asked us to find the value of 𝑥, which forms part of the expression for 𝐵𝐷. Let’s think about how to approach this problem.
The line 𝐴𝐷 is a bisector of the angle 𝐶𝐴𝐵. 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 𝐶𝐵 into two parts, 𝐶𝐷 and 𝐷𝐵.
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 𝐵𝐷 by 𝐶𝐷 is the same as the ratio we get when we divide 𝐴𝐵 by 𝐴𝐶. In each case, this is the pink side divided by the green side.
We can substitute in the values or, in the case of 𝐵𝐷, the expression for each of these sides to give an equation that we can solve to find the value of 𝑥. 𝐵𝐷 over 𝐶𝐷 becomes 𝑥 plus 10 over 12. 𝐴𝐵 over 𝐴𝐶 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 let’s 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 we’re left with 𝑥 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 𝑥 is equal to four. So we found the value of 𝑥. 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.