Question

Given: △ABC, m∠A=120°, AB=AC=1 Find: The radius of circumscribed circle

Answer 1

In Δ ABC, ∠A=120°, AB=AC=1

To draw a circumscribed circle Draw perpendicular bisectors of any of two sides.The point where these bisectors meet is the center of the circle.Mark the center as O.

Then join OA, OB, and OC.

Taking any one OA,OB and OC as radius draw the circumcircle.

Now, from O Draw OM⊥AB and ON⊥AC.

As chord AB and AC are equal,So OM and ON will also be equal.

The reason being that equal chords are equidistant from the center.

AM=MB=1/2 and AN=NC=1/2  [ perpendicular from the center to the chord bisects the chord.]

In Δ OMA and ΔONA

OM=ON [proved above]

OA is common.

MA=NA=1/2  [proved above]

ΔOMA≅ ONA [SSS]

∴ ∠OAN =∠OAM=60° [ CPCT]

In Δ OAN

$\cos60=\frac {AN}{OA}$

$\frac{1}{2}=\frac{\frac{1}{2}}{OA}$

OA=1

∴ OA=OB=OC=1, which is the radius of given Circumscribed circle.