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Test: Triangle lines and circles

This test develops understanding of key triangle lines, including angle bisectors, medians and altitudes. It also covers incircles and circumcircles, their centres and related length calculations.

Question qty: 10
Duration in minutes: 19
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Sample questions

  • The sides are a cm, b cm and c cm, and area is s cm². Find the circumradius using S = abc/(4R).
  • One angle of a cyclic quadrilateral is a°. Find the opposite angle.
  • A triangle has area s cm² and semiperimeter p cm. Find its inradius.
  • An angle bisector divides the opposite side into x cm and X cm. The adjacent sides are a cm and b cm. Find X.
  • Which statement precisely describes an angle bisector of a triangle?
  • Three consecutive sides of a tangential quadrilateral are a cm, b cm and c cm. Find the fourth.
  • The inradius is r cm and semiperimeter is p cm. Find the area.
  • Find the midpoint of the segment with endpoints (x1, y1) and (x2, y2).

Triangle lines and circles

The medians meet at the centroid, which divides every median in the ratio 2 : 1 from the vertex. Angle bisectors meet at the incentre, while perpendicular bisectors of sides meet at the circumcentre.

Altitudes and radii

The altitudes meet at the orthocentre. For the incircle, area A = rs where s is the semiperimeter; for the circumcircle, A = abc/(4R). Identify each centre from the defining lines rather than from its apparent position in a diagram.

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