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In a triangle ABC, take a point D on BC such that DB = 14, DA = 13, DC=4 and the circum circle of ADB is congruent to the circum circle of ADC. What is the area of triangle ABC?

In a triangle ABC, take a point D on BC such that DB = 14, DA = 13, DC=4 and the circum circle of ADB is congruent to the circum circle of ADC. What is the area of triangle ABC?

Grade:10

2 Answers

Vikas TU
14149 Points
3 years ago
The fact that the two circumcircles are congruent means that the chord AD must subtend the same angle in both circles.
That is, ∠ABC = ∠ACB, so ABC is isosceles.
Drop the perpendicular M from A to BC;
we know MC = 9 and so MD = 5
and by Pythagoras on AMD, AM = 12.
Therefore, the area of ABC is
1/2 (AM)(BC) = 1/2 (12)(18) = 108.
satwika
13 Points
4 months ago
The fact that the two circumcircles are congruent means that the chord AD must subtend the same angle in both circles.
That is, ∠ABC = ∠ACB, so ABC is isosceles.
Drop the perpendicular M from A to BC;
we know MC = 9 and so MD = 5
and by Pythagoras on AMD, AM = 12.
Therefore, the area of ABC is
1/2 (AM)(BC) = 1/2 (12)(18) = 108

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