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Let G be the centroid of DABC in which the angle C is obtuse. Let AD and CF are the medians from A and C on the sides BC and AB respectively. If the four points B, D, G and F are concyclic, then BC/AC
- a)> 1/2
- b)< 1/√2
- c)> 1/√2
- d)< 1/2
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
Let G be the centroid of DABC in which the angle C is obtuse. Let AD a...
using power of point for the circle w.r.t. point ‘C’
CD.CB = CG.CF
a2 + c2 = 2b2
also, ‘C’ is obtuse ⇒ cosC < 0
a2 + b2 < c2
⇒ a2 + b2 < 2b2 – a2
⇒ 2a2 < b2
CD.CB = CG.CF
a2 + c2 = 2b2
also, ‘C’ is obtuse ⇒ cosC < 0
a2 + b2 < c2
⇒ a2 + b2 < 2b2 – a2
⇒ 2a2 < b2
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Let G be the centroid of DABC in which the angle C is obtuse. Let AD a...
Let's consider the triangle ABC with centroid G.
Since G is the centroid, we know that AG = 2GD and CG = 2GF.
Let's denote the intersection point of AD and CF as point P.
Since AD is a median, we have BP = PC.
Also, since CF is a median, we have AP = 2PD.
Now, let's consider the circle passing through points B, D, G, and F.
Since B, D, G, and F are concyclic, we can denote the center of this circle as O.
Since O is the center of the circle, we know that OB = OD = OG = OF.
Now, let's consider the triangles BOC and AOC.
Since OB = OC, the triangles BOC and AOC are congruent by the side-side-side (SSS) congruence criterion.
Therefore, angle BCO = angle ACO.
Since angle ACO = angle ACF (since CF is a median), we have angle BCO = angle ACF.
Similarly, angle BOC = angle ADC (since AD is a median).
Since angle BCO = angle ACF and angle BOC = angle ADC, we have angle ACF = angle ADC.
Therefore, triangle ACF is an isosceles triangle with base AF.
Since angle C is obtuse, we have angle ACF > angle CAF (since the angles of a triangle add up to 180 degrees).
Therefore, angle ACF > angle CAF > angle AFC.
Since triangle AFC is an isosceles triangle, we have angle AFC = angle ACF.
Therefore, angle AFC > angle ACF > angle AFC, which is a contradiction.
Therefore, the assumption that B, D, G, and F are concyclic is false.
Hence, BC/AC is not defined.
Since G is the centroid, we know that AG = 2GD and CG = 2GF.
Let's denote the intersection point of AD and CF as point P.
Since AD is a median, we have BP = PC.
Also, since CF is a median, we have AP = 2PD.
Now, let's consider the circle passing through points B, D, G, and F.
Since B, D, G, and F are concyclic, we can denote the center of this circle as O.
Since O is the center of the circle, we know that OB = OD = OG = OF.
Now, let's consider the triangles BOC and AOC.
Since OB = OC, the triangles BOC and AOC are congruent by the side-side-side (SSS) congruence criterion.
Therefore, angle BCO = angle ACO.
Since angle ACO = angle ACF (since CF is a median), we have angle BCO = angle ACF.
Similarly, angle BOC = angle ADC (since AD is a median).
Since angle BCO = angle ACF and angle BOC = angle ADC, we have angle ACF = angle ADC.
Therefore, triangle ACF is an isosceles triangle with base AF.
Since angle C is obtuse, we have angle ACF > angle CAF (since the angles of a triangle add up to 180 degrees).
Therefore, angle ACF > angle CAF > angle AFC.
Since triangle AFC is an isosceles triangle, we have angle AFC = angle ACF.
Therefore, angle AFC > angle ACF > angle AFC, which is a contradiction.
Therefore, the assumption that B, D, G, and F are concyclic is false.
Hence, BC/AC is not defined.
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Let G be the centroid of DABC in which the angle C is obtuse. Let AD and CF are the medians from A and C on the sides BC and AB respectively. If the four points B, D, G and F are concyclic, then BC/ACa)> 1/2b)< 1/√2c)> 1/√2d)< 1/2Correct answer is option 'B'. Can you explain this answer? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Let G be the centroid of DABC in which the angle C is obtuse. Let AD and CF are the medians from A and C on the sides BC and AB respectively. If the four points B, D, G and F are concyclic, then BC/ACa)> 1/2b)< 1/√2c)> 1/√2d)< 1/2Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let G be the centroid of DABC in which the angle C is obtuse. Let AD and CF are the medians from A and C on the sides BC and AB respectively. If the four points B, D, G and F are concyclic, then BC/ACa)> 1/2b)< 1/√2c)> 1/√2d)< 1/2Correct answer is option 'B'. Can you explain this answer?.
Let G be the centroid of DABC in which the angle C is obtuse. Let AD and CF are the medians from A and C on the sides BC and AB respectively. If the four points B, D, G and F are concyclic, then BC/ACa)> 1/2b)< 1/√2c)> 1/√2d)< 1/2Correct answer is option 'B'. Can you explain this answer? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Let G be the centroid of DABC in which the angle C is obtuse. Let AD and CF are the medians from A and C on the sides BC and AB respectively. If the four points B, D, G and F are concyclic, then BC/ACa)> 1/2b)< 1/√2c)> 1/√2d)< 1/2Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let G be the centroid of DABC in which the angle C is obtuse. Let AD and CF are the medians from A and C on the sides BC and AB respectively. If the four points B, D, G and F are concyclic, then BC/ACa)> 1/2b)< 1/√2c)> 1/√2d)< 1/2Correct answer is option 'B'. Can you explain this answer?.
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