In triangle ABC, the bisector of angle BAC meets BC at point D in such a way that AB = 10 cm, AC = 15 cm and BD = 6 cm. Find the length of BC (in cm).
Show Hint
Always remember the property \( \frac{\text{Left Side}}{\text{Right Side}} = \frac{\text{Left Segment}}{\text{Right Segment}} \) for internal angle bisectors in any triangle.
Step 1: Understanding the Question:
The problem describes a triangle ABC where the angle bisector of angle A meets side BC at a point D.
We are given the lengths of sides AB, AC, and segment BD, and we need to determine the total length of the base side BC. Step 2: Key Formula or Approach:
This problem is solved using the Angle Bisector Theorem.
The Angle Bisector Theorem states that the angle bisector of an interior angle of a triangle divides the opposite side into two segments that are proportional to the adjacent sides of the triangle.
The formula is:
\[ \frac{AB}{AC} = \frac{BD}{CD} \] Step 3: Detailed Explanation:
1. We are given the following values:
\( AB = 10 \text{ cm} \)
\( AC = 15 \text{ cm} \)
\( BD = 6 \text{ cm} \)
2. Let the length of segment \( CD \) be denoted as \( x \text{ cm} \).
3. According to the Angle Bisector Theorem, the ratio of the side lengths adjacent to the bisected angle is equal to the ratio of the segments created on the opposite side:
\[ \frac{AB}{AC} = \frac{BD}{CD} \]
4. Substituting the given values into the ratio:
\[ \frac{10}{15} = \frac{6}{x} \]
5. Simplify the fraction on the left side:
\[ \frac{2}{3} = \frac{6}{x} \]
6. Cross-multiply to solve for \( x \):
\[ 2x = 3 \times 6 \]
\[ 2x = 18 \]
\[ x = 9 \text{ cm} \]
7. This gives us the length of the segment \( CD = 9 \text{ cm} \).
8. The total length of the side \( BC \) is the sum of the segments \( BD \) and \( CD \) because point D lies on the line segment BC:
\[ BC = BD + CD \]
9. Substituting the segment values:
\[ BC = 6 + 9 = 15 \text{ cm} \]
Step 4: Final Answer:
The length of BC is 15 cm.