Math Problem Statement

Find the value of x given a cyclic quadrilateral inside a circle where arcs RS and ST are equal in length.

Solution

The diagram shows a circle with four points PP, QQ, RR, and SS on the circumference. We are tasked with finding the value of angle xx, where the arcs RSRS and STST are equal in length.

Let's break this down:

Step 1: Identify the angles in the cyclic quadrilateral

Since the diagram shows a cyclic quadrilateral (a quadrilateral inscribed in a circle), we can use the property that the opposite angles of a cyclic quadrilateral add up to 180180^\circ. This means: P+R=180\angle P + \angle R = 180^\circ Given:

  • R=130\angle R = 130^\circ
  • Let P=x\angle P = x

Thus: x+130=180x + 130^\circ = 180^\circ Solving this gives: x=180130=50x = 180^\circ - 130^\circ = 50^\circ

Therefore, the value of xx is 5050^\circ.

Summary:

The angle xx is 5050^\circ.

Would you like further details or have any questions about the steps? Here are some additional questions to explore this topic:

  1. What other properties of cyclic quadrilaterals can help solve problems like this?
  2. How do we know arcs RSRS and STST being equal affect the solution?
  3. Can we find any other angles in the diagram using similar reasoning?
  4. What would change if arcs RSRS and STST were not equal?
  5. How would this problem differ if the circle were not given?

Tip: Remember that the sum of opposite angles in a cyclic quadrilateral always equals 180180^\circ. This is a key property when dealing with circle theorems!

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Math Problem Analysis

Mathematical Concepts

Geometry
Circle Theorems
Cyclic Quadrilateral

Formulas

Sum of opposite angles of a cyclic quadrilateral = 180°

Theorems

Cyclic Quadrilateral Theorem

Suitable Grade Level

Grades 8-10