Finding the Length of BC in an Isosceles Triangle with an Angle Bisector

In the isosceles triangle ABC, where angle A is 100° and sides AB and AC are equal, the internal angle bisector BD has a length of 20 units. This article explains how to calculate the length of side BC using the angle bisector theorem and trigonometric relationships.

Step 1: Determine the Angles

Since AB AC, triangle ABC is isosceles. Therefore, we can find angles B and C:

angle B angle C 80° / 2 40°

Step 2: Use the Angle Bisector Theorem

The angle bisector BD divides side AC into two segments AD and DC. By the angle bisector theorem:

AD/DC AB/BC

If AB AC c and BC a, then:

AD c * a / (c - a) and DC a * a / (c - a)

Step 3: Calculate the Length of BC Using the Angle Bisector Length Formula

The formula for the length of an angle bisector BD in triangle ABC is given by:

BD (2 * AB * AC) / (AB AC) * cos(A/2)

Substituting AB AC c and angle A 100°:

BD (2 * c2) / (2c) * cos(50°) c * cos(50°)

Step 4: Express c in Terms of a

Using the cosine of 50°:

cos(50°) sin(40°)

Using the identity sin(40°) sin(90° - 50°) cos(50°), we can express c:

20 c * sin(50°)

Using sin(50°) cos(40°):

sin(50°) sin(90° - 40°) cos(40°)

Using sin(10°) 0.174, we have:

sin(50°) sin(40°) * cos(10°)

Step 5: Using the Law of Sines

We can relate a to c using the Law of Sines:

a / sin(100°) c / sin(40°)

Thus,

a c * (sin(100°) / sin(40°))

Step 6: Calculate c

Now we can substitute back to find c:

BD 20 c * sin(50°)

Using the approximation sin(50°) ≈ 0.766:

c 20 / sin(50°) ≈ 20 / 0.766 ≈ 26.1

Step 7: Calculate BC

Now substituting back to find a:

a c * (sin(100°) / sin(40°)) 26.1 * (0.9848 / 0.6428)

a ≈ 26.1 * 1.533 ≈ 40.1

Conclusion

Rounding to the nearest integer, the length of BC is:

boxed{40}