Question

find the value of x.
2
30°
5
5
30°
x
x = ?

Explanation:

Step1: Apply the law of cosines

For the left - hand triangle with sides \(a = 2\), \(b = 5\), and included angle \(\theta=30^{\circ}\), the law of cosines states \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\). For the right - hand triangle with sides \(a'=x\), \(b' = 5\), and included angle \(\theta'=30^{\circ}\), since the two triangles are congruent (by SAS - Side - Angle - Side congruence criterion as two sides and the included angle are equal in both triangles: the radii of the circle are equal and the angles are equal), we can also use the law of cosines. But we can also use the property of congruent triangles directly.

Step2: Use triangle congruence

Since the two triangles have two sides equal (the radii of the circle are both 5) and the included angles equal (\(30^{\circ}\)), the two triangles are congruent. So, if one non - radius side is 2, the other non - radius side \(x\) must be equal to 2.

Answer:

2