Math Problem Statement

Solve for class 9 cbse

Find the value of x, given angles at point O in a cyclic quadrilateral.

Solution

Let's analyze the image and solve the problem based on the information given.

Problem Breakdown:

The problem asks to find the value of xx. The figure shows a cyclic quadrilateral with certain angles marked.

  1. Key Information:
    • AOB=135\angle AOB = 135^\circ
    • AOE=20\angle AOE = 20^\circ
    • BOC=4x\angle BOC = 4x
    • EOC=2x\angle EOC = 2x

Approach:

  1. Use of Angle Sum Property in a Circle: For a quadrilateral inscribed in a circle, opposite angles sum to 180180^\circ.

  2. Angle Relationship at Point OO: Since AOBEAOBE forms a quadrilateral, and OO is the center, we can use the properties of angles subtended by arcs and solve for xx.

Step-by-step Solution:

  1. Angle at Point OO: Sum of all angles at point O=360\text{Sum of all angles at point O} = 360^\circ Thus, AOB+AOE+BOC+EOC=360\angle AOB + \angle AOE + \angle BOC + \angle EOC = 360^\circ Plugging in the given values: 135+20+4x+2x=360135^\circ + 20^\circ + 4x + 2x = 360^\circ Simplifying: 155+6x=360155^\circ + 6x = 360^\circ 6x=3601556x = 360^\circ - 155^\circ 6x=2056x = 205^\circ x=2056x = \frac{205^\circ}{6} x=34.17x = 34.17^\circ

Answer:

The value of xx is approximately 34.17°.

Would you like more details or have further questions?

Here are some related questions to expand on the topic:

  1. What are the properties of a cyclic quadrilateral?
  2. How can you calculate the angles subtended by an arc in a circle?
  3. What is the significance of opposite angles in a cyclic quadrilateral?
  4. Can this type of problem appear with different configurations of angles?
  5. How does the sum of angles around a point work in geometrical figures?

Tip: Always double-check your angle calculations, especially when dealing with multiple unknowns in a geometric figure.

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Math Problem Analysis

Mathematical Concepts

Cyclic Quadrilateral
Angle Sum Property
Geometry

Formulas

Sum of angles around a point = 360°
Opposite angles in a cyclic quadrilateral sum to 180°

Theorems

Angle Sum Property
Cyclic Quadrilateral Theorem

Suitable Grade Level

Grade 9