Angle Bisector Theorem Explained:

The Angle Bisector Theorem is a geometric principle that involves angle bisectors in triangles. An angle bisector is a line or ray that divides an angle into two equal parts.

Statement of the Angle Bisector Theorem:

If an angle bisector of a triangle divides the opposite side into two segments, then the ratio of the lengths of these two segments is equal to the ratio of the lengths of the other two sides of the triangle.

Mathematically, if in triangle $\triangle ABC$, the angle bisector of $\angle A$ intersects side $BC$ at point $D$, the theorem says:

$\frac{BD}{DC} = \frac{AB}{AC}$

Where:

• $\angle A$ is the angle being bisected.
• $AB$ and $AC$ are the two sides of the triangle.
• $BD$ and $DC$ are the two segments created on side $BC$.

Why It Matters:

This theorem is helpful because it gives a direct relationship between the sides of a triangle when we know that an angle bisector is present. It’s commonly used in problems involving triangle proportions, geometric constructions, and proofs.

Interactive Example:

Let’s solve an example step-by-step to understand this better.

Given:

• Triangle $ABC$ where the angle bisector of $\angle A$ intersects side $BC$ at point $D$.
• $AB = 8$, $AC = 6$, and $BD = 4$.

We need to find the length of $DC$ (the other segment on side $BC$).

Using the Angle Bisector Theorem:

From the Angle Bisector Theorem, we know: $\frac{BD}{DC} = \frac{AB}{AC}$

Substitute the known values: $\frac{4}{DC} = \frac{8}{6}$

Simplify the right-hand side: $\frac{4}{DC} = \frac{4}{3}$

Now, cross-multiply: $4 \times 3 = 4 \times DC$ $12 = 4 \times DC$

Solve for $DC$: $DC = \frac{12}{4} = 3$

So, $DC = 3$.

Conclusion:

The length of segment $DC$ is 3 units.

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Do you want to see another example or further explore how the theorem is applied in different problems?

5 Questions for Further Learning:

1. How do you find the angle bisector in a triangle without using a protractor?
2. Can the Angle Bisector Theorem be applied to all types of triangles, including right-angled ones?
3. How would the theorem change if the triangle was not scalene?
4. What other geometric theorems are used along with the Angle Bisector Theorem in solving real-world problems?
5. How can you use the Angle Bisector Theorem to prove that a point is equidistant from two sides of a triangle?

Tip:

Always check that the angle bisector is correctly identified in a diagram—misinterpreting the angle bisector can lead to incorrect application of the theorem!