All questions of Triangles for SSC CGL Exam

Understanding the Problem
In triangle ABC, we have the angle bisector AD, with given lengths AB = 3 cm and AC = 1 cm. We need to find the ratio BD : BC.
Angle Bisector Theorem
The Angle Bisector Theorem states that the ratio of the lengths of the two segments created by the angle bisector on the opposite side is equal to the ratio of the other two sides of the triangle. In this case:
- Let BD = x
- Let DC = y
According to the theorem:
x / y = AB / AC
Applying the Theorem
Substituting the known values:
- AB = 3 cm
- AC = 1 cm
This gives us:
x / y = 3 / 1
Expressing BD and BC
Now, since BD + DC = BC, we can express:
BC = x + y
To find the ratio BD : BC, we can substitute y in terms of x:
y = (1/3)x
Then, substituting this back into the expression for BC gives:
BC = x + (1/3)x = (4/3)x
Finding the Ratio
Now we find the ratio:
BD : BC = x : (4/3)x
This simplifies to:
BD : BC = 1 : (4/3)
To express it in a more conventional format:
BD : BC = 3 : 4
Conclusion
Thus, the final ratio of BD to BC is:
Correct answer: option 'D' (3 : 4).

Given Data:
- Length of side AB = 12 cm
- Length of side BC = 8 cm
- Angle C = 59°

Calculating Side AC using Law of Cosines:
- Using Law of Cosines, we have the formula: c² = a² + b² - 2ab * cos(C)
- Substituting the given values, we get: AC² = 12² + 8² - 2 * 12 * 8 * cos(59°)
- Calculating the cosine value of 59°, we get: cos(59°) ≈ 0.5299
- Plugging in the values, we get: AC² = 144 + 64 - 192 * 0.5299
- Simplifying further, we get: AC² = 144 + 64 - 101.8368
- Therefore, AC² ≈ 106.1632
- Taking the square root of both sides, we get: AC ≈ √106.1632 ≈ 10.305

Final Answer:
Therefore, the length of side AC is approximately 10.3 cm. Hence, option 'C' (14) is the correct answer.