Solving Triangle Problems with Angle Bisector Theorem

This problem involves finding the length of a side in a triangle where an angle bisector is present. The key concept to use here is the Angle Bisector Theorem.

The Angle Bisector Theorem states that if a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle.

In Δ ABC, AD is the angle bisector of ∠ A, intersecting side BC at point D. According to the Angle Bisector Theorem, the ratio of the length of side AB to the length of side AC (or CA) is equal to the ratio of the length of the segment BD to the length of the segment DC.

Mathematically, this can be written as:

\(\frac{\text{AB}}{\text{AC}} = \frac{\text{BD}}{\text{DC}}\)

Applying the Angle Bisector Theorem to find CA

We are given the following lengths:

• AB = 12 cm
• BD = 7 cm
• DC = 8.75 cm

We need to find the length of CA. Let's substitute the given values into the Angle Bisector Theorem formula:

\(\frac{12}{\text{CA}} = \frac{7}{8.75}\)

Now, we can solve for CA by cross-multiplication:

\(12 \times 8.75 = \text{CA} \times 7\)

Calculate the product on the left side:

\(12 \times 8.75 = 105\)

So, the equation becomes:

\(105 = 7 \times \text{CA}\)

To find CA, divide 105 by 7:

\(\text{CA} = \frac{105}{7}\)

\(\text{CA} = 15\)

Therefore, the length of CA is 15 cm.

Calculation Summary

The length of CA is 15 cm.

Revision Table: Triangle Geometry & Theorems

Additional Information: Properties of Triangles

Understanding various triangle properties and theorems is crucial for solving geometry problems. Beyond the Angle Bisector Theorem, here are a few other important concepts:

• Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This helps determine if a triangle can be formed with given side lengths.
• Pythagorean Theorem: In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (\(a^2 + b^2 = c^2\)).
• Median: A line segment joining a vertex to the midpoint of the opposite side.
• Altitude: A perpendicular line segment from a vertex to the opposite side (or its extension).
• Centroid, Orthocenter, Circumcenter, Incenter: These are special points of concurrency formed by medians, altitudes, perpendicular bisectors, and angle bisectors, respectively.

Each of these concepts has specific properties and theorems associated with them, which are useful in different types of triangle problems.