Circle Geometry Practice Questions with Detailed Solutions – Level 3

Circle Geometry Level 3 Practice Questions on Chords, Tangents and Radii

Circle Geometry Level 3 is an advanced quantitative aptitude practice set on chords, tangents, secants, common tangents, cyclic quadrilaterals and the inradius and circumradius of triangles. On Aptitude360 every question comes with a worked solution. The problems combine more than one circle property, so they reward clear diagrams and careful reasoning.

Topics in this Level 3 circle geometry set

• Parallel chords on the same or opposite sides of the centre
• Intersecting chords and chords through an interior point
• Tangent and secant lengths from an external point
• Direct and transverse common tangents of two circles
• Angles in cyclic quadrilaterals and the alternate segment theorem
• Annulus area between concentric circles
• Inradius and circumradius of triangles, and common chords of intersecting circles

Key circle formulas and rules to revise

• Intersecting chords: \( PA\times PB=PC\times PD \)
• Tangent and secant: \( PT^{2}=PA\times PB \)
• Direct common tangent: \( \sqrt{d^{2}-(r_{1}-r_{2})^{2}} \)
• Transverse common tangent: \( \sqrt{d^{2}-(r_{1}+r_{2})^{2}} \)
• Chord at distance \( d \): \( 2\sqrt{r^{2}-d^{2}} \)
• Circumradius: \( R=\dfrac{abc}{4\Delta} \)

Preparation tips for hard circle problems

• Draw the perpendicular from the centre to a chord, since it bisects the chord and creates a right triangle.
• Join the centre to the point of contact, as this radius is perpendicular to the tangent.
• For triangle radii, find the area with Heron's formula first, then divide by the semi-perimeter for the inradius.
• In cyclic quadrilaterals, use the fact that opposite angles are supplementary before chasing other angles.

Who this circle geometry set suits

Aptitude students who are comfortable with basic circle theorems and want harder geometry practice for competitive exams.