Question

48. ABC is a triangle with \(\angle CAB = 15^{\circ}\) and \(\angle ABC = 30^{\circ}\). If M is the midpoint of AB, then \(\angle ACM = ?\)

• A. \(15^{\circ}\)
• B. \(30^{\circ}\)
• C. \(45^{\circ}\)
• D. \(60^{\circ}\)

Hint:

Find angle ACB using the angle sum property, then split it in the ratio of the two base angles since CM goes to the midpoint of AB.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
M is the midpoint of side AB, so CM is the median from C. A useful property of this median is that it divides the vertex angle at C into two parts that are proportional to the two base angles of the triangle.

Step 2: Key Formula or Approach:
First find the full angle at C using the angle sum property, then split it in the ratio of the base angles $\angle ABC : \angle CAB$.

Step 3: Detailed Explanation:
The angle sum of triangle ABC gives:
\[ \angle ACB = 180^{\circ} - \angle CAB - \angle ABC = 180^{\circ} - 15^{\circ} - 30^{\circ} = 135^{\circ} \]
Let $\angle ACM = x$ and $\angle BCM = 135^{\circ} - x$. Since the median splits the angle in the ratio of the base angles:
\[ \frac{x}{135^{\circ} - x} = \frac{\angle CAB}{\angle ABC} = \frac{15^{\circ}}{30^{\circ}} = \frac{1}{2} \]
Cross multiplying:
\[ 2x = 135^{\circ} - x \implies 3x = 135^{\circ} \implies x = 45^{\circ} \]

Step 4: Final Answer:
So $\angle ACM = 45^{\circ}$, which is the required angle.