Question

what is the value of x? x - 51° x =

Explanation:

Step1: Recall circle - angle property

In a circle, if two chords are equal, the arcs they subtend are equal and the angles subtended by the equal - length chords at the circumference are equal. Here, since the two chords are equal (indicated by the red marks), the triangle $\triangle VUW$ is isosceles. So, the base - angles are equal. Let the base - angles be $\angle VUW=\angle UVW$. And we know that the sum of angles in a triangle is $180^{\circ}$. Also, if we assume the center of the circle is $O$, and the triangle is inscribed in the circle. Let's consider the angle - sum property of a triangle.

Step2: Set up an equation

In $\triangle VUW$, if one of the non - equal angles is $x - 51^{\circ}$, and the other two equal angles are such that the sum of all three angles is $180^{\circ}$. Since the triangle is isosceles, we have $2(x - 51^{\circ})+90^{\circ}=180^{\circ}$ (assuming the triangle is a right - angled isosceles triangle inscribed in a semi - circle, as one of the chords passes through the center). First, expand the left - hand side: $2x-102^{\circ}+90^{\circ}=180^{\circ}$. Then simplify the left - hand side to get $2x - 12^{\circ}=180^{\circ}$.

Step3: Solve the equation for $x$

Add $12^{\circ}$ to both sides of the equation: $2x=180^{\circ}+12^{\circ}=192^{\circ}$. Then divide both sides by 2: $x = 96^{\circ}$.

Answer:

$96$