Question

(A) The diagram below shows a circle, with the points P. O. R and S lying on its circumference and its center marked O. RP is a diameter of the circle and AB is a tangent to the circle at P. Angle APQ=3x°, angle QPR = 2x°. RPS = x° and angleOSP=54° . Figure below here_Determine the value of EACH of the following angles. Show detailed working where possible and give a reason for your answer. (i) x (ii) y (iii) =

Answer 1

The question lacks a visual representation of the geometric figure, which is essential for calculating the requested angles. Without the diagram, we cannot provide the values of angles x, y, and z.

Step-by-step explanation:

The student's question pertains to finding the values of angles in a geometric figure involving a circle, a diameter, a tangent, and various marked points and angles. To solve such problems, we often utilize properties of circles, tangents, and angles.

Unfortunately, without a visual representation of the figure, the specific calculation of angles x, y, and z cannot be completed, as those angles depend on the geometric relationships within the figure. Normally, we would use the property that the angle formed by a tangent and a radius is 90 degrees, and that the angles in a quadrilateral add up to 360 degrees, among others, to solve for x, y, and z.

However, to assist the student in a meaningful way, further clarification or a diagram from the student would be necessary.

Answer 2

The angle subtended by a diameter/semicircle on any point on the circumference of a circle is 90°.

Therefore, ∠RSP = 90°

The Tangent-Chord Theorem, states that: The angle measure between a chord of a circle and a tangent through any of the chord's endpoints is equal to the measure of an angle in the alternate segment.

Using the Tangent-Chord Theorem, it follows that:

`\angle APQ=\angle PSQ`

Notice that:

`\angle APQ=3x`

and

`\angle PSQ=54^(\circ)`

Therfore, we have that:

`3x=54^(\circ)`

Divide both sides of the equation by 3:

`\begin{gathered} (3x)/(3)=(54^(\circ))/(3) \\ x=18^(\circ) \end{gathered}`

Notice that the interior angles of triangle PRS are:

`\angle RSP,x,\text{ and },y`

Since the sum of the interior angles of a triangle is 180°.

Therefore, we must have that:

The Same Segment Theorem states that the angles at the circumference subtended by the same arc are equal. More simply, angles in the same segment are equal.

Since angles y and z are both subtended by arc SP, it follows from the Same Segment Theorem that:

`\begin{gathered} z=y \\ \text{ Since }y=72^(\circ),\text{ it follows that:} \\ z=72^(\circ) \end{gathered}`

Therefore, the answer is:

(i) x = 18°, (ii)y = 72°, and (iii) z = 72°