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Let s denote the semiperimeter of a triangle ABC in which BC = a, CA = b, AB = c. If a circle touches the sides BC, CA, AB at D, E, F, respectively, find BD.
- a)s – b
- b)2s + h
- c)b + s
- d)3b – s
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
Let s denote the semiperimeter of a triangle ABC in which BC = a, CA =...
By the property of a circle tangent to a line, we know that the line segment from the center of the circle to the point of tangency is perpendicular to the line. Therefore, let O be the center of the circle and let M, N, and P be the midpoints of BC, CA, and AB respectively.
Since O is the incenter of triangle ABC, O is equidistant from the sides of the triangle. Therefore, OM = ON = OP.
We can draw radii from O to the points of tangency D, E, and F. Let r be the length of these radii. Then, OD = OE = OF = r.
Since OM = ON = OP, we have that MD = NE = PF = s - a, where s is the semiperimeter of triangle ABC.
By the Pythagorean Theorem, we have that BD^2 = BM^2 + MD^2. Since BM = a/2 and MD = s - a, we have that BD^2 = (a/2)^2 + (s - a)^2.
Expanding and simplifying, we have that BD^2 = a^2/4 + s^2 - 2as + a^2.
Rearranging terms, we have that BD^2 = s^2 - 2as + a^2/4 + a^2.
Factoring, we have that BD^2 = (s - a/2)^2 + a^2/4.
Taking the square root of both sides, we have that BD = sqrt((s - a/2)^2 + a^2/4).
Therefore, the length of BD is sqrt((s - a/2)^2 + a^2/4).
Answer: a) sqrt((s - a/2)^2 + a^2/4)
Since O is the incenter of triangle ABC, O is equidistant from the sides of the triangle. Therefore, OM = ON = OP.
We can draw radii from O to the points of tangency D, E, and F. Let r be the length of these radii. Then, OD = OE = OF = r.
Since OM = ON = OP, we have that MD = NE = PF = s - a, where s is the semiperimeter of triangle ABC.
By the Pythagorean Theorem, we have that BD^2 = BM^2 + MD^2. Since BM = a/2 and MD = s - a, we have that BD^2 = (a/2)^2 + (s - a)^2.
Expanding and simplifying, we have that BD^2 = a^2/4 + s^2 - 2as + a^2.
Rearranging terms, we have that BD^2 = s^2 - 2as + a^2/4 + a^2.
Factoring, we have that BD^2 = (s - a/2)^2 + a^2/4.
Taking the square root of both sides, we have that BD = sqrt((s - a/2)^2 + a^2/4).
Therefore, the length of BD is sqrt((s - a/2)^2 + a^2/4).
Answer: a) sqrt((s - a/2)^2 + a^2/4)
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Let s denote the semiperimeter of a triangle ABC in which BC = a, CA =...
According to question,
s = (a + b + c/2) ⇒ 2s = a + b + c
B is external point and BD and BF are tangents and from an external point the tangents drawn to a circle are equal in length.
∴ BD = BF;
Similarly, AF = AE; CD = CE
s = Semi perimeter =
⇒ 2s = AB + AC + BC
⇒ 2s = AF + FB + AE + EC + BD + DC
⇒ 2s = 2AE + 2CE + 2BD
⇒ s = AE + CE + BD
⇒ s = AC + BD ⇒ s - b = BD.
s = (a + b + c/2) ⇒ 2s = a + b + c
B is external point and BD and BF are tangents and from an external point the tangents drawn to a circle are equal in length.
∴ BD = BF;
Similarly, AF = AE; CD = CE
s = Semi perimeter =
⇒ 2s = AB + AC + BC
⇒ 2s = AF + FB + AE + EC + BD + DC
⇒ 2s = 2AE + 2CE + 2BD
⇒ s = AE + CE + BD
⇒ s = AC + BD ⇒ s - b = BD.
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Let s denote the semiperimeter of a triangle ABC in which BC = a, CA = b, AB = c. If a circle touches the sides BC, CA, AB at D, E, F, respectively,find BD.a)s – bb)2s + hc)b + sd)3b – sCorrect answer is option 'A'. 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 s denote the semiperimeter of a triangle ABC in which BC = a, CA = b, AB = c. If a circle touches the sides BC, CA, AB at D, E, F, respectively,find BD.a)s – bb)2s + hc)b + sd)3b – sCorrect answer is option 'A'. 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 s denote the semiperimeter of a triangle ABC in which BC = a, CA = b, AB = c. If a circle touches the sides BC, CA, AB at D, E, F, respectively,find BD.a)s – bb)2s + hc)b + sd)3b – sCorrect answer is option 'A'. Can you explain this answer?.
Let s denote the semiperimeter of a triangle ABC in which BC = a, CA = b, AB = c. If a circle touches the sides BC, CA, AB at D, E, F, respectively,find BD.a)s – bb)2s + hc)b + sd)3b – sCorrect answer is option 'A'. 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 s denote the semiperimeter of a triangle ABC in which BC = a, CA = b, AB = c. If a circle touches the sides BC, CA, AB at D, E, F, respectively,find BD.a)s – bb)2s + hc)b + sd)3b – sCorrect answer is option 'A'. 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 s denote the semiperimeter of a triangle ABC in which BC = a, CA = b, AB = c. If a circle touches the sides BC, CA, AB at D, E, F, respectively,find BD.a)s – bb)2s + hc)b + sd)3b – sCorrect answer is option 'A'. Can you explain this answer?.
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