# Find $\angle B$ for Triangle $ABC$ with Bisectors $AD,BE$

• MHB
• maxkor
In summary: AB = \frac{AE^2-BC^2}{AE+BC}$. In summary, we can use the Angle Bisector Theorem to find the lengths of$AD$and$BE$. Then, by setting up an equation and using the Law of Cosines, we can find the length of$AB$in terms of$B$. Finally, we can use the Law of Cosines again to find the measure of$B$. Therefore, the measure of$B$is$\frac{AE^2-BC^2}{AE+BC}$. maxkor Let$ABC$be a triangle with$\angle A= 60^{\circ},$and$AD,BE$are bisectors of$A,B$respectively where$D\in BC, E\in AC.$Find the measure of$B$if$AB+BD=AE+BE.$Hello, thank you for your question. First, we can use the Angle Bisector Theorem to find the lengths of$AD$and$BE$. Since$AD$and$BE$are bisectors, we know that$\frac{BD}{DC} = \frac{AB}{AC}$and$\frac{BE}{AE} = \frac{AB}{BC}$. Since$\angle A = 60^{\circ}$, we can use the fact that the angles in a triangle add up to$180^{\circ}$to find that$\angle B = 90^{\circ}$. Next, we can use the given information to set up an equation. Since$AB+BD=AE+BE$, we can substitute in the lengths we found using the Angle Bisector Theorem to get$\frac{AB \cdot BC}{AC} + \frac{AB \cdot AC}{BC} = \frac{AB \cdot BC}{AE} + \frac{AB \cdot AE}{BC}$. Simplifying this equation, we get$AB^2 = AE \cdot AC$. Using the Law of Cosines, we can find the length of$AB$in terms of$B$. Since$\angle A = 60^{\circ}$, we have$AB^2 = AC^2 + BC^2 - 2AC \cdot BC \cdot \cos(60^{\circ})$. Simplifying this, we get$AB^2 = AC^2 + BC^2 - AC \cdot BC$. Now, we can substitute this into our previous equation to get$AC^2 + BC^2 - AC \cdot BC = AE \cdot AC$. Factoring out$AC$, we get$AC(AC-BC) = AE \cdot AC$. Since$AC \neq 0$, we can divide both sides by$AC$to get$AC-BC = AE$. Finally, we can use the Law of Cosines again to find the measure of$B$. Since$\angle A = 60^{\circ}$, we have$AC^2 = AB^2 + BC^2 - 2AB \cdot BC \cdot \cos(60^{\circ})$. Substituting in the lengths we found earlier, we get$AE^2 = AB^2 + BC^2 - AB \cdot BC$. Simplifying this, we get$

## 1. What are bisectors and how are they related to angles in a triangle?

Bisectors are lines that divide an angle into two equal parts. In a triangle, the bisectors of each angle intersect at a point called the incenter. This point is equidistant from the sides of the triangle and is the center of the inscribed circle.

## 2. How do I find the measure of an angle using bisectors in a triangle?

To find the measure of an angle using bisectors, you can use the angle bisector theorem. This states that the angle bisector of an angle in a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle. By setting up and solving a proportion, you can find the measure of the angle.

## 3. Can I use bisectors to find the measure of any angle in a triangle?

Yes, you can use bisectors to find the measure of any angle in a triangle. This is because the angle bisector theorem applies to all angles in a triangle, not just the ones that are bisected.

## 4. How many bisectors are there in a triangle?

There are three bisectors in a triangle, one for each angle. These bisectors intersect at the incenter of the triangle.

## 5. Can bisectors be used to find the area of a triangle?

No, bisectors cannot be used to find the area of a triangle. They only help in finding the measure of angles in a triangle. To find the area of a triangle, you need to know the length of at least one side and the height of the triangle.

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