Consider the following statements 1 If non parallel sides of a trapezium are equal then it is cyclic...........

Quantitative Aptitude (CDS) Coordinate Geometry

Consider the following statements:

1. If non-parallel sides of a trapezium are equal, then it is cyclic.

2. If the chord of a circle is equal to its radius, then the angle subtended by this chord at a point in major segment is 30°.

Which of the above statements is/are correct?

1 only

2 only

Both 1 and 2

Neither 1 nor 2

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Explanation:

In ΔAPD and ΔBQC,
AD = BC (Given)
DP = CQ (Distance between parallel sides)
= ∠APD ∠BQC (90°)
∴ ΔAPD ≅ ΔBQC (RHS congruence rule)
⇒ ∠A = ∠B (CPCT)
∠A + ∠D = 180° (Sum of adjacent interior angles is supplementary)
⇒ ∠B + ∠D = 180°
Therefore, trapezium ABCD is cyclic.
Hence, statement (1) is correct.

In ΔOAB, OA = OB = AB = r
ΔOAB is an equilateral triangle.
⇒ ∠AOB = 60°
Now,∠AOB = 2∠ACB (Degreemeasure theorem)
∴ 2∠ACB=60°
⇒ ∠ACB = 30°
Hence, statement (2) is correct.