Solving Triangle Side Length using Angle Bisector Theorem

The question asks us to find the length of side CA in triangle ABC, given the length of AB, the lengths of the segments BD and DC created by the angle bisector AD, and the fact that AD bisects angle A internally.

We are given:

• AB = 10 cm
• BD = 6 cm
• DC = 7.5 cm
• AD is the internal bisector of ∠ A, intersecting BC at D.

We need to find the length of CA.

Understanding the Angle Bisector Theorem

The Angle Bisector Theorem is a fundamental concept in geometry that relates the lengths of the sides of a triangle to the lengths of the segments created by an angle bisector of a vertex.

The theorem states that if an angle of a triangle is bisected by a line that intersects the opposite side, then the ratio of the lengths of the other two sides of the triangle is equal to the ratio of the lengths of the two segments created on the opposite side.

For triangle ABC, where AD is the angle bisector of ∠ A meeting BC at D, the theorem can be stated as:

\[ \frac{\text{AB}}{\text{AC}} = \frac{\text{BD}}{\text{DC}} \]

or equivalently,

\[ \frac{\text{AB}}{\text{BD}} = \frac{\text{AC}}{\text{DC}} \]

Applying the Angle Bisector Theorem to Find CA

We can use the relationship given by the Angle Bisector Theorem to find the unknown length CA. We have the values for AB, BD, and DC.

Let's substitute the given values into the formula:

\[ \frac{\text{AB}}{\text{AC}} = \frac{\text{BD}}{\text{DC}} \]

\[ \frac{10 \text{ cm}}{\text{CA}} = \frac{6 \text{ cm}}{7.5 \text{ cm}} \]

Now, we need to solve this equation for CA. We can do this by cross-multiplication:

\[ 10 \times 7.5 = \text{CA} \times 6 \]

Calculate the product on the left side:

\[ 75 = 6 \times \text{CA} \]

To isolate CA, divide both sides of the equation by 6:

\[ \text{CA} = \frac{75}{6} \]

Now, we simplify the fraction:

\[ \text{CA} = \frac{25 \times 3}{2 \times 3} \]

Cancel out the common factor of 3:

\[ \text{CA} = \frac{25}{2} \]

Convert the fraction to a decimal:

\[ \text{CA} = 12.5 \text{ cm} \]

Thus, the length of side CA is 12.5 cm.

Calculation Summary

The calculated length of CA is 12.5 cm.

Revision Table: Angle Bisector Theorem

Additional Information: Related Geometric Concepts

Understanding the Angle Bisector Theorem is key to solving this problem. Here are some related concepts:

• Triangle Properties: Triangles have various properties related to their sides, angles, medians, altitudes, and angle bisectors.
• External Angle Bisector Theorem: Similar to the internal angle bisector theorem, the external angle bisector of a triangle also relates the sides and segments on the extended opposite side. For Δ ABC, if the external bisector of ∠ A meets the extension of BC at E, then \( \frac{\text{AB}}{\text{AC}} = \frac{\text{BE}}{\text{CE}} \).
• Similarity of Triangles: While not directly used here, the Angle Bisector Theorem can sometimes be proven using similar triangles or trigonometric approaches.
• Ceva's Theorem: This theorem deals with concurrent lines (like angle bisectors, medians, altitudes) from the vertices of a triangle to the opposite sides. If AD, BE, CF are concurrent cevians in Δ ABC, then \( \frac{\text{BD}}{\text{DC}} \times \frac{\text{CE}}{\text{EA}} \times \frac{\text{AF}}{\text{FB}} = 1 \).

These theorems help in solving various geometry problems involving triangles and their specific lines.