Question

Given below are two statements
Statement I =If two chords XY and ZT of a circle intersects internally at point P,then PX-PYPZ-PT
Statement II =If two chords XY and ZT of a circle intersect internally at point P,then PXZ and PTY are similar triangles
In the light of the above statements, choose the correct answer from the options given below

• A. Both Statement I and Statement II are true
• B. Both Statement i and Statement II are false
• C. Statement I is true but Statement II is false
• D. Statement I is false but Statement II is true

Answer 1

The Correct Option is A

To solve this problem, we need to analyze the two statements given in the context of circle geometry.

Statement I:

If two chords \(XY\) and \(ZT\) of a circle intersect internally at point \(P\), then PX \cdot PY = PZ \cdot PT.

This statement corresponds to the properties of intersecting chords in circle geometry. When two chords intersect inside a circle, the product of the lengths of the segments of one chord is equal to the product of the segments of the other chord. This is a well-known theorem in circle geometry, often referred to as the chord intersection theorem. Therefore, Statement I is true.

Statement II:

If two chords \(XY\) and \(ZT\) of a circle intersect internally at point \(P\), then triangles \(PXZ\) and \(PTY\) are similar.

To verify this, note that when two chords intersect, the opposite angles formed are equal (vertically opposite angles). Thus, \(\angle XPZ = \angle YPT\). Also, \(\angle PZX = \angle PTY\) as they subtend the same arcs on the circle. Hence, by AA (Angle-Angle) similarity criterion, triangles \(PXZ\) and \(PTY\) are similar. Therefore, Statement II is also true.

Conclusion:

Both Statement I and Statement II are true. The correct answer is Both Statement I and Statement II are true.

Tips:

• Remember the chord intersection theorem for quick verification of segment products.
• Use angle properties and similarity criteria to determine relationships between triangles in circle problems.