We are given a triangle $ABC$ with side lengths $AB = 7$, $BC = 6$, and $CA = 8$. The angle bisectors $CP$ and $CQ$ are drawn at vertex $C$, where $CP$ is the interior angle bisector and $CQ$ is the exterior angle bisector. We need to find the lengths of segments $AP$, $BP$, $AQ$, and $PQ$.

Step 1: Use the Angle Bisector Theorem for $AP$ and $BP$

For the interior angle bisector $CP$, we apply the Angle Bisector Theorem, which states that the interior angle bisector divides the opposite side into segments proportional to the adjacent sides. Thus, we have:

$\frac{AP}{BP} = \frac{AC}{BC} = \frac{8}{6} = \frac{4}{3}$

Let $AP = 4x$ and $BP = 3x$. Since $AP + BP = AB = 7$, we can set up the following equation:

$4x + 3x = 7$

Solving for $x$:

$7x = 7 \implies x = 1$

Thus, the lengths of $AP$ and $BP$ are:

$AP = 4 \times 1 = 4 \quad \text{and} \quad BP = 3 \times 1 = 3$

Step 2: Use the Extended Angle Bisector Theorem for $AQ$ and $BQ$

For the exterior angle bisector $CQ$, we use the Extended Angle Bisector Theorem, which states that the exterior angle bisector divides the opposite side into segments such that:

$\frac{AQ}{BQ} = \frac{AC}{BC} = \frac{8}{6} = \frac{4}{3}$

Let $AQ = 4y$ and $BQ = 3y$. Since $AQ - BQ = AB = 7$ for the exterior angle bisector, we set up the following equation:

$4y - 3y = 7$

Solving for $y$:

$y = 7$

Thus, the lengths of $AQ$ and $BQ$ are:

$AQ = 4 \times 7 = 28 \quad \text{and} \quad BQ = 3 \times 7 = 21$

Step 3: Find the length of $PQ$

Since $P$ and $Q$ lie on opposite sides of $BC$, the length of $PQ$ can be found using the formula:

$PQ = \frac{2 \sqrt{AB \cdot AC \cdot BC \cdot s}}{AB + AC}$

where $s$ is the semi-perimeter of the triangle. Let me calculate that for you next.The length of segment $PQ$ is approximately $7.92$.

Final Results:

• $AP = 4$
• $BP = 3$
• $AQ = 28$
• $PQ \approx 7.92$

Would you like further details or have any questions about this process?

Here are 5 related questions to extend your understanding:

1. How does the Angle Bisector Theorem apply to other types of triangles (e.g., isosceles)?
2. What is the relationship between interior and exterior angle bisectors in a triangle?
3. How do you derive the formula for the length of segment $PQ$ in this context?
4. Can you use this method to find the lengths of segments if one of the sides is unknown?
5. How does this problem change if the triangle is not scalene?

Tip: The angle bisector theorem holds regardless of the type of triangle (scalene, isosceles, etc.), as long as you're bisecting the angle.