Question

In triangle ABC, AB = AC = x, \( \angle ABC = \theta \) and the circumradius is equal to y. Then \( \frac{x}{y} \) equals

• A. \( 2 \cos \theta \)
• B. \( 2 \sin \theta \)
• C. \( \sin \theta \)
• D. \( \cos \theta \)

Hint:

The Sine Rule is the fundamental relationship between sides, angles, and the circumradius of a triangle. Whenever a problem involves the circumradius, the Sine Rule should be the first tool you consider.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem connects the sides and angles of a triangle to its circumradius using the Sine Rule.
Step 2: Key Formula or Approach:
The Sine Rule states that for any triangle with sides a, b, c and opposite angles A, B, C, and circumradius R: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \] Step 3: Detailed Explanation:
In triangle ABC, we are given:

AB = c = x
AC = b = x
\( \angle ABC = B = \theta \)
Circumradius R = y
Since AB = AC, the triangle is isosceles. Therefore, the angles opposite to these sides are equal: \( \angle ACB = C = \angle ABC = \theta \).
We can apply the Sine Rule using the side-angle pair we know. Let's use side AC (length b=x) and its opposite angle B (\( \theta \)). According to the Sine Rule: \[ \frac{b}{\sin B} = 2R \] Substituting the given values: \[ \frac{x}{\sin \theta} = 2y \] The question asks for the value of \( \frac{x}{y} \). We can rearrange the equation to find this ratio. \[ \frac{x}{y} = 2 \sin \theta \] Step 4: Final Answer:
The value of \( \frac{x}{y} \) is \( 2 \sin \theta \).