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?
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evilcat:
what type of triangle ABC is?
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see the diagram
let O be the center of the circle
therefore, AE is the diameter
AE = AD + DE
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
please mark the best
let O be the center of the circle
therefore, AE is the diameter
AE = AD + DE
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
please mark the best
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