Question Video: Using the Angle Bisector Theorem to Find Unknown Terms Mathematics

Given that π΄π΅πΆ is a triangle in which π΄πΆ = 10 cm, find the value of each of π₯ and π¦.

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Video Transcript

Given that π΄π΅πΆ is a triangle in which π΄πΆ equals 10 centimeters, find the value of each of π₯ and π¦.

Letβs have a look at the figure, in which we can observe the triangle π΄π΅πΆ and also record the fact that π΄πΆ is equal to 10 centimeters. Letβs see what else we can observe in this figure. Well, we can also see that one of the exterior angles of triangle π΄π΅πΆ has been bisected. The whole of the angle πΈπ΄πΆ has been split into two congruent angles such that angle πΈπ΄π· is congruent to angle π·π΄πΆ.

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 π·πΆ over π·π΅ is equal to π΄πΆ over π΄π΅. And we can then substitute in the given lengths for these sides. On the left-hand side, we have that the length of π·πΆ is π¦ plus four. However, π·π΅ will be equal to six plus π¦ plus four, which is π¦ plus 10. Then, on the right-hand side, we know that π΄πΆ is 10 and π΄π΅ 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 π¦ plus four is equal to two times π¦ plus 10. Then, by distributing the three and two across our parentheses, we have three π¦ plus 12 is equal to two π¦ plus 20. We then collect the like terms, which we can do in one or two steps, which leaves us with π¦ is equal to eight. This means that we have found out the value of one of the unknown variables of π¦. So letβs clear some space so that we can work out the value of π₯.

We can identify that π₯ lies on the line segment π΄π·. It is this line segment π΄π· which is in fact the exterior angle bisector. Because the line segment π΄π· bisects the angle πΈπ΄πΆ, which is the external angle to angle π΅π΄πΆ, and it intersects the line π΅πΆ at the point π·, then we can determine the length of the line segment π΄π·. To find the length of the exterior angle bisector in this figure, we have that π΄π· is equal to the square root of π΅π· times π·πΆ minus π΄π΅ times π΄πΆ.

When we go to substitute in the lengths, we begin with the length of the line segment π΅π·. As we previously calculated that π¦ is equal to eight, then π΅π· is six plus eight plus four, which is 18 centimeters. Then, we have that π·πΆ must be 12 centimeters, π΄π΅ is 15 centimeters, and π΄πΆ is 10 centimeters. So we have π΄π· 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 π₯ is root 66.

As we werenβt given any indication in the question that we need to give our answer as a decimal, then we can keep this value of π₯ in the root form. And therefore, by using the fact that triangle π΄π΅πΆ has an exterior angle which has been bisected, we have found that the value of π₯ is root 66 and the value of π¦ is equal to eight.