# The bisector of angle BAC of triangle ABC cuts line BC at D and circumcircle of the triangle at point E. If DE is 3 cm & AC is 4 cm. AD is 5 cm. Find AB? 15 10 40 20

1
by nickiey
what type of triangle ABC is?

2015-03-16T19:24:43+05:30

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See the diagram
let O be the center of the circle
therefore, AE is the diameter
AE = 5 + 3
AE = 8 cm
therefore,
AO = r
AO = 8 ÷ 2
AO = 4 cm
In triangle AOC
AO = OC = 4 cm                 (radii)
AO = OC = AC = 4 cm            (given AC = 4cm)
therefore,
Triangle AOC is an equilateral traingle

In triangle AOB
AO = OB = 4 cm                      (radii)
therefore,
angle OAB = angle OBA          (angles opp. to equal sides are equal)
Therefore
angle OBA = angle OAB = angle OAC = angle OCA    (angle OAB = angle                                                  OAC given, angle OAB = angle OBA, angle OAC                                              = angle OCA)
In traingles AOB and AOC
AO is common
angle OAB = angle OAC          (given)
angle OBA = angle OCA          (proved above)
therefore by AAS congruence rule
Triangle AOB = Triangle AOC
therefore by CPCT
AB = AC
AB = 4 cm