Question 34 of 50beginner🔧 ApplyNumerical3 marks

Quadrilateral PQRS is inscribed in a circle. If ∠\angleP = (2x+10)∘(2x + 10)^\circ and ∠\angleR = (3x−20)∘(3x - 20)^\circ, find the value of xx and the measures of ∠\angleP and ∠\angleR.

Correct Answer

x = 38, ∠P = 86°, ∠R = 94°

Exercise: END-OF-CHAPTER EXERCISES | Q: 8 | (Page 23)
For More Understanding

Explanation

The problem applies the property of a cyclic quadrilateral, which states that the sum of its opposite angles is 180°. Since PQRS is inscribed in a circle, ∠P and ∠R are opposite angles. Equating their sum to 180° allows us to solve for x, and subsequently find the individual angle measures.

Solution Steps

  1. Step 1: In a cyclic quadrilateral, the sum of opposite pairs of angles is 180°. Therefore, ∠P + ∠R = 180°.

  2. Step 2: Substitute the given expressions: (2x + 10) + (3x - 20) = 180.

  3. Step 3: Simplify the equation: 5x - 10 = 180, which gives 5x = 190, so x = 38.

  4. Step 4: Calculate ∠P = (2×382 \times 38) + 10 = 76 + 10 = 86°.

  5. Step 5: Calculate ∠R = (3×383 \times 38) - 20 = 114 - 20 = 94°.

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