Angle Bisector Theorem: Practice Problems

📐 Understanding the Angle Bisector Theorem

The Angle Bisector Theorem states that given triangle $\triangle ABC$ and angle bisector $AD$, where $D$ is a point on $BC$, then the ratio of $AB$ to $AC$ is equal to the ratio of $BD$ to $DC$. Mathematically, this is expressed as:

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

✍️ Practice Problem 1

In $\triangle ABC$, $AB = 8$, $AC = 6$, and $BC = 7$. If $AD$ bisects $\angle BAC$, find the length of $BD$.

Solution:

1. Let $BD = x$. Then $DC = BC - BD = 7 - x$.
2. Apply the Angle Bisector Theorem: $\frac{AB}{AC} = \frac{BD}{DC}$
3. Substitute the given values: $\frac{8}{6} = \frac{x}{7-x}$
4. Cross-multiply: $8(7-x) = 6x$
5. Simplify: $56 - 8x = 6x$
6. Combine like terms: $56 = 14x$
7. Solve for $x$: $x = \frac{56}{14} = 4$

Therefore, $BD = 4$.

✍️ Practice Problem 2

In $\triangle PQR$, $PQ = 12$, $PR = 18$, and $QR = 15$. If $PS$ bisects $\angle QPR$, find the length of $QS$.

Solution:

1. Let $QS = x$. Then $SR = QR - QS = 15 - x$.
2. Apply the Angle Bisector Theorem: $\frac{PQ}{PR} = \frac{QS}{SR}$
3. Substitute the given values: $\frac{12}{18} = \frac{x}{15-x}$
4. Simplify the fraction: $\frac{2}{3} = \frac{x}{15-x}$
5. Cross-multiply: $2(15-x) = 3x$
6. Simplify: $30 - 2x = 3x$
7. Combine like terms: $30 = 5x$
8. Solve for $x$: $x = \frac{30}{5} = 6$

Therefore, $QS = 6$.

✍️ Practice Problem 3

In $\triangle XYZ$, $XY = 10$, $YZ = 9$, and $XZ = 7$. If $XW$ bisects $\angle YXZ$, find the length of $YW$.

Solution:

1. Let $YW = x$. Then $WZ = YZ - YW = 9 - x$.
2. Apply the Angle Bisector Theorem: $\frac{XY}{XZ} = \frac{YW}{WZ}$
3. Substitute the given values: $\frac{10}{7} = \frac{x}{9-x}$
4. Cross-multiply: $10(9-x) = 7x$
5. Simplify: $90 - 10x = 7x$
6. Combine like terms: $90 = 17x$
7. Solve for $x$: $x = \frac{90}{17} \approx 5.29$

Therefore, $YW \approx 5.29$.

💻 Code Verification (Python)

Here's a Python function to verify the Angle Bisector Theorem:

def angle_bisector_theorem(side1, side2, base, segment1):
    segment2 = base - segment1
    return abs((side1 / side2) - (segment1 / segment2)) < 1e-6  # Using tolerance for floating-point comparison

# Example usage (Problem 1):
side1 = 8
side2 = 6
base = 7
segment1 = 4

print(angle_bisector_theorem(side1, side2, base, segment1))

🧠 Key Takeaways

• The Angle Bisector Theorem provides a relationship between the sides of a triangle and the segments created by an angle bisector.
• It's a powerful tool for solving geometric problems involving triangles.
• Remember to set up the ratios correctly: $\frac{AB}{AC} = \frac{BD}{DC}$.