Question

AC is a tangent to the circle below at point B.

Calculate the sizes of angle x, angle y and angle z.
Justify each of your answers.​

Answer 1

To calculate angles x, y, and z, acknowledge that a tangent creates a 90° angle with the circle's radius, use trigonometric functions if side lengths are provided, and apply the sum of angles in a triangle.

Step-by-step explanation:

To solve for the angles x, y, and z given in the problem, we must remember that a tangent to a circle creates a right angle with the radius at the point of tangency. Additionally, if we consider the triangle formed, we can utilize trigonometric relationships. Without the specific diagram referenced, a general solution can't be provided, but the steps to solve for the angles would involve the following:

• Recognize that angle x, formed by the tangent and the radius, is 90° due to the right angle property of tangents to circles.
• If given the lengths of the triangle sides, use trigonometric ratios (sine, cosine, tangent) to determine the other angles.
• Apply the sum of angles in a triangle, which is 180°, to find the remaining angle if two are known.

Once angle x is known to be 90°, the other angles can be found using properties of triangles and circle theorems. For instance, if the central angle is given, angle y can be deduced since the angle at the centre is twice the angle at the circumference. Again, without the visual diagram, specific values for angles y and z cannot be calculated.

Answer 2

`y=78^(\circ)` (alternate interior angles theorem)

`x=78^(\circ)` (alternate segment theorem)

`z=24^(\circ)` (angles in a triangle add to 180 degrees)