Understanding the Triangle Angle Bisector Problem

This problem asks us to find the length of a segment on one side of a triangle created by an angle bisector from the opposite vertex. We are given the lengths of all three sides of triangle ABC, and we know that BD is the bisector of angle B, meeting side AC at point D.

Applying the Angle Bisector Theorem

To solve this problem, we need to use a fundamental concept in geometry known as the Angle Bisector Theorem. This theorem provides a relationship between the lengths of the sides of a triangle and the segments created by an angle bisector on the opposite side.

The Angle Bisector Theorem states that in a triangle, the angle bisector of any angle divides the opposite side in the ratio of the lengths of the other two sides. For ΔABC, where BD is the angle bisector of ∠B meeting AC at D, the theorem says:

$\frac{AD}{DC} = \frac{AB}{BC}$

Calculating the Length of AD

We are given the following lengths:

• AB = 12 cm
• BC = 18 cm
• AC = 15 cm

According to the Angle Bisector Theorem, we have:

$\frac{AD}{DC} = \frac{AB}{BC}$

Substitute the given values for AB and BC:

$\frac{AD}{DC} = \frac{12}{18}$

Simplify the fraction $\frac{12}{18}$:

$\frac{12}{18} = \frac{6 \times 2}{6 \times 3} = \frac{2}{3}$

So, the ratio $\frac{AD}{DC} = \frac{2}{3}$. This means that the segment AD and the segment DC are in the ratio 2 : 3. The side AC is divided into two segments, AD and DC, and their sum is equal to the length of AC.

AD + DC = AC

AD + DC = 15 cm

Since AD : DC = 2 : 3, we can consider AC as being divided into $2 + 3 = 5$ parts. The length of AD corresponds to 2 of these parts, and the length of DC corresponds to 3 of these parts.

To find the length of AD, we can use the ratio in relation to the total length of AC:

$AD = \left(\frac{\text{Ratio part for AD}}{\text{Total ratio parts}}\right) \times AC$

$AD = \left(\frac{2}{2 + 3}\right) \times 15$

$AD = \left(\frac{2}{5}\right) \times 15$

Now, calculate the value:

$AD = \frac{2 \times 15}{5}$

$AD = \frac{30}{5}$

$AD = 6$

Therefore, the length of AD is 6 cm.

We can also find the length of DC to verify: $DC = \left(\frac{\text{Ratio part for DC}}{\text{Total ratio parts}}\right) \times AC$ $DC = \left(\frac{3}{2 + 3}\right) \times 15$ $DC = \left(\frac{3}{5}\right) \times 15$ $DC = \frac{3 \times 15}{5} = \frac{45}{5} = 9$ cm.

Check: AD + DC = 6 cm + 9 cm = 15 cm, which is indeed the length of AC.

Conclusion

Using the Angle Bisector Theorem and the given side lengths of triangle ABC, we found that the bisector of angle B divides the side AC into segments AD and DC in the ratio 2:3. By dividing the total length of AC (15 cm) according to this ratio, we determined the length of segment AD.

The length of AD is 6 cm.

Revision Table: Key Concepts for Triangle Angle Bisector

Additional Information: Related Geometry Concepts

Understanding the Angle Bisector Theorem is crucial for solving various geometry problems involving triangles. Here are some related concepts:

• Triangle Properties: Triangles have many interesting properties related to their sides, angles, medians, altitudes, and angle bisectors.
• Medians: A median connects a vertex to the midpoint of the opposite side. Medians intersect at the centroid.
• Altitudes: An altitude is a perpendicular segment from a vertex to the opposite side. Altitudes intersect at the orthocenter.
• Perpendicular Bisectors: A line segment perpendicular to a side and passing through its midpoint. Perpendicular bisectors intersect at the circumcenter (center of the circumscribed circle).
• Incenter: The point where the three angle bisectors of a triangle intersect. The incenter is the center of the inscribed circle (incircle). The distance from the incenter to each side is equal (radius of the incircle).
• Other Triangle Theorems: Besides the Angle Bisector Theorem, other important theorems include the Pythagorean theorem, Sine Rule, Cosine Rule, and theorems related to similar and congruent triangles.

Mastering these concepts helps build a strong foundation in geometry.