We are given the following diagram and asked to find the value of variables x and y.
First, we can find the value of x. To do this we will use the second part of the Inscribed Right Triangle Theorem, which states the following.
Inscribed Right Triangle Theorem
If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.
From the diagram we can see that XZ passes trough the center of the circle, so it is a diameter of the circle. Hence, by this theorem, angle ∠ Y opposite to XZ is a right angle. This allows us to conclude that 3x equals 90.
3x=90
⇕
x=30
Now we can find the value of y. Let's recall that by the Triangle Sum Theorem the sum of the measures of the interior angles of a triangle is 180^(∘). Therefore, the following is true.
34^(∘)+3x^(∘)+2y^(∘)=180^(∘)
Let's substitute x with 30 and solve this equation for y.
