Math Problem Statement

Solve

Solution

The problem shows a circle with center OO, and asks to find the value of x+yx + y. The given conditions are:

  • PQS=40\angle PQS = 40^\circ
  • RSP=75\angle RSP = 75^\circ

I'll walk through the steps to solve this geometrically.

Step-by-Step Solution:

  1. Identifying the Angles:

    • PQS=40\angle PQS = 40^\circ (this is an angle at the circumference).
    • RSP=75\angle RSP = 75^\circ (another angle at the circumference).
  2. Understanding the Circle Theorems:

    • The angles subtended by the same arc at the circumference of a circle are equal.
    • The sum of the opposite angles in a cyclic quadrilateral is 180180^\circ.
  3. Finding yy:

    • Notice that yy is subtended by the same arc as PQS\angle PQS.
    • Therefore, by the circle theorem, y=PQS=40y = \angle PQS = 40^\circ.
  4. Finding xx:

    • xx and RSP\angle RSP are opposite angles in a cyclic quadrilateral, meaning their sum should be 180180^\circ.
    • Therefore, x+RSP=180x + \angle RSP = 180^\circ.
    • Substituting RSP=75\angle RSP = 75^\circ, we get: x+75=180x + 75^\circ = 180^\circ x=18075=105x = 180^\circ - 75^\circ = 105^\circ
  5. Calculating x+yx + y:

    • Now, x=105x = 105^\circ and y=40y = 40^\circ.
    • So, x+y=105+40=145x + y = 105^\circ + 40^\circ = 145^\circ.

Final Answer:

The value of x+yx + y is 145145^\circ.


Do you have any questions about this solution? Would you like further clarification?

Here are some related questions that might help you explore this topic further:

  1. What are the key properties of cyclic quadrilaterals?
  2. How do you use circle theorems to solve geometric problems?
  3. What is the relationship between the angles subtended by the same arc in a circle?
  4. How would you solve this problem if the diagram involved a different set of angles?
  5. What is the significance of opposite angles in cyclic quadrilaterals?

Tip: When working with circle geometry, always check for opportunities to apply circle theorems, as they often simplify the problem significantly.

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Math Problem Analysis

Mathematical Concepts

Geometry
Circle Geometry
Circle Theorems

Formulas

-

Theorems

Circle theorems

Suitable Grade Level

Grades 7-9