Given that 𝐴𝐵𝐶 is a triangle in which 𝐴𝐶 equals 10 centimeters, find the value of each of 𝑥 and 𝑦.
Let’s have a look at the figure, in which we can observe the triangle 𝐴𝐵𝐶 and also record the fact that 𝐴𝐶 is equal to 10 centimeters. Let’s see what else we can observe in this figure. Well, we can also see that one of the exterior angles of triangle 𝐴𝐵𝐶 has been bisected. The whole of the angle 𝐸𝐴𝐶 has been split into two congruent angles such that angle 𝐸𝐴𝐷 is congruent to angle 𝐷𝐴𝐶.
Because it is an exterior angle which has been bisected, then that means that we can apply the exterior angle bisector theorem. By this theorem, we can write that the ratio of 𝐷𝐶 over 𝐷𝐵 is equal to 𝐴𝐶 over 𝐴𝐵. And we can then substitute in the given lengths for these sides. On the left-hand side, we have that the length of 𝐷𝐶 is 𝑦 plus four. However, 𝐷𝐵 will be equal to six plus 𝑦 plus four, which is 𝑦 plus 10. Then, on the right-hand side, we know that 𝐴𝐶 is 10 and 𝐴𝐵 is 15. Before we cross multiply, we can take out a common factor of five on the right-hand side to give us two-thirds.
So then, when we cross multiply, we have three times 𝑦 plus four is equal to two times 𝑦 plus 10. Then, by distributing the three and two across our parentheses, we have three 𝑦 plus 12 is equal to two 𝑦 plus 20. We then collect the like terms, which we can do in one or two steps, which leaves us with 𝑦 is equal to eight. This means that we have found out the value of one of the unknown variables of 𝑦. So let’s clear some space so that we can work out the value of 𝑥.
We can identify that 𝑥 lies on the line segment 𝐴𝐷. It is this line segment 𝐴𝐷 which is in fact the exterior angle bisector. Because the line segment 𝐴𝐷 bisects the angle 𝐸𝐴𝐶, which is the external angle to angle 𝐵𝐴𝐶, and it intersects the line 𝐵𝐶 at the point 𝐷, then we can determine the length of the line segment 𝐴𝐷. To find the length of the exterior angle bisector in this figure, we have that 𝐴𝐷 is equal to the square root of 𝐵𝐷 times 𝐷𝐶 minus 𝐴𝐵 times 𝐴𝐶.
When we go to substitute in the lengths, we begin with the length of the line segment 𝐵𝐷. As we previously calculated that 𝑦 is equal to eight, then 𝐵𝐷 is six plus eight plus four, which is 18 centimeters. Then, we have that 𝐷𝐶 must be 12 centimeters, 𝐴𝐵 is 15 centimeters, and 𝐴𝐶 is 10 centimeters. So we have 𝐴𝐷 is equal to the square root of 18 times 12 minus 15 times 10. Simplifying this gives the square root of 216 minus 150, which simplifies to the square root of 66. So we have determined that the exterior angle bisector has a length of root 66 centimeters. This means that we can say that the value of 𝑥 is root 66.
As we weren’t given any indication in the question that we need to give our answer as a decimal, then we can keep this value of 𝑥 in the root form. And therefore, by using the fact that triangle 𝐴𝐵𝐶 has an exterior angle which has been bisected, we have found that the value of 𝑥 is root 66 and the value of 𝑦 is equal to eight.