Understanding the Triangle Geometry Problem

The question asks us to find the length of side CA in triangle ABC. We are given the length of side AB and how the angle bisector of angle A divides the opposite side BC into segments BD and DC. This scenario is a direct application of a fundamental geometric theorem.

Applying the Angle Bisector Theorem

The key to solving this problem is the Angle Bisector Theorem. This theorem states that if a line bisects an angle of a triangle and intersects the opposite side, then it divides the opposite side into segments that are proportional to the other two sides of the triangle.

In our triangle ΔABC, AD is the internal bisector of ∠A and it intersects BC at D. According to the Angle Bisector Theorem, the ratio of the lengths of the two sides containing the angle (∠A) must be equal to the ratio of the lengths of the two segments the angle bisector divides the opposite side (BC) into. That is:

$$\frac{\text{AB}}{\text{CA}} = \frac{\text{BD}}{\text{DC}}$$

Given Values in ΔABC

We are provided with the following lengths:

• Length of side AB = 8 cm
• Length of segment BD = 6 cm
• Length of segment DC = 7.5 cm

We need to find the length of side CA.

Calculating the Length of CA

Now, we substitute the given values into the Angle Bisector Theorem equation:

$$\frac{8}{\text{CA}} = \frac{6}{7.5}$$

To find CA, we can rearrange the equation. We can cross-multiply:

$$8 \times 7.5 = \text{CA} \times 6$$

Now, we solve for CA:

$$\text{CA} = \frac{8 \times 7.5}{6}$$

Let's perform the multiplication and division:

$$\text{CA} = \frac{60}{6}$$

$$\text{CA} = 10$$

So, the length of side CA is 10 cm.

Step-by-Step Solution Summary

Here are the steps taken to find the length of CA:

1. Identify the given information: AB, BD, and DC.
2. Recognize that AD is an angle bisector and apply the Angle Bisector Theorem.
3. Write down the ratio relationship: $\frac{\text{AB}}{\text{CA}} = \frac{\text{BD}}{\text{DC}}$.
4. Substitute the known values: $\frac{8}{\text{CA}} = \frac{6}{7.5}$.
5. Solve the equation for CA.
6. Calculate the final value of CA.

The calculated length of CA is 10 cm.

Revision Table: Key Triangle Concepts

Additional Information: Understanding Angle Bisectors

An angle bisector in a triangle has another important property: any point on the angle bisector is equidistant from the two sides that form the angle. For example, any point on AD is the same distance from side AB as it is from side AC.

The Angle Bisector Theorem is a powerful tool in geometry problems involving triangles and angle bisectors. It's crucial to correctly identify the sides and the segments of the opposite side when applying the theorem.