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

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 $\frac{AE^2-BC^2}{AE+BC}$.
  • #1
maxkor
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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 $AB+BD=AE+BE.$
 
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  • #2


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 $
 

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

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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