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A circle of radius 4cm is inscribed in ΔABC, which touches the side BC at D, if BD = 6cm and DC = 8cm, then which of the following options are correct ?
- a)The triangle is necessarily acute angled triangle
- b)tan A/2 = 4/7
- c)Perimeter of the ΔABC is 42cm
- d)Area of ΔABC is 84 sq cm
Correct answer is option 'A,B,C,D'. Can you explain this answer?
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Most Upvoted Answer
A circle of radius 4cm is inscribed in ΔABC, which touches the side B...
Given information:
- A circle with radius 4cm is inscribed in triangle ABC.
- The circle touches side BC at point D.
- BD = 6cm and DC = 8cm.
Analysis:
To solve this problem, we need to examine the properties of the inscribed circle and the given triangle.
1. Angle BAC:
Let's consider angle BAC. Angle BAC is the angle opposite to side BC. Since the circle is inscribed in triangle ABC and touches side BC at point D, the line segment AD is the angle bisector of angle BAC. Therefore, angle BAD is equal to angle CAD.
2. Triangle Inequality Theorem:
According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In triangle ABC, we have:
- AB + AC > BC
- AB + BC > AC
- AC + BC > AB
Proof:
1. Angle BAC:
Since the circle is inscribed in triangle ABC and touches side BC at point D, the line segment AD is the angle bisector of angle BAC. Therefore, angle BAD is equal to angle CAD.
2. Triangle Inequality Theorem:
According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In triangle ABC, we have:
- AB + AC > BC
- AB + BC > AC
- AC + BC > AB
Conclusion:
- The triangle is necessarily acute angled triangle because the sum of the angles in a triangle is always 180 degrees and the obtuse angle can't be formed in this case.
- tan A/2 = 4/7 because angle BAD is equal to angle CAD and angle BAD is half of angle BAC.
- The perimeter of triangle ABC is 42cm because AB + BC + AC = 6 + 8 + 28 = 42.
- The area of triangle ABC is 84 sq cm because the area of a triangle can be calculated using the formula: Area = (1/2) * base * height. In this case, the base is BC and the height is the radius of the inscribed circle. Therefore, the area is (1/2) * 14 * 4 = 84 sq cm.
Therefore, options A, B, C, and D are correct.
- A circle with radius 4cm is inscribed in triangle ABC.
- The circle touches side BC at point D.
- BD = 6cm and DC = 8cm.
Analysis:
To solve this problem, we need to examine the properties of the inscribed circle and the given triangle.
1. Angle BAC:
Let's consider angle BAC. Angle BAC is the angle opposite to side BC. Since the circle is inscribed in triangle ABC and touches side BC at point D, the line segment AD is the angle bisector of angle BAC. Therefore, angle BAD is equal to angle CAD.
2. Triangle Inequality Theorem:
According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In triangle ABC, we have:
- AB + AC > BC
- AB + BC > AC
- AC + BC > AB
Proof:
1. Angle BAC:
Since the circle is inscribed in triangle ABC and touches side BC at point D, the line segment AD is the angle bisector of angle BAC. Therefore, angle BAD is equal to angle CAD.
2. Triangle Inequality Theorem:
According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In triangle ABC, we have:
- AB + AC > BC
- AB + BC > AC
- AC + BC > AB
Conclusion:
- The triangle is necessarily acute angled triangle because the sum of the angles in a triangle is always 180 degrees and the obtuse angle can't be formed in this case.
- tan A/2 = 4/7 because angle BAD is equal to angle CAD and angle BAD is half of angle BAC.
- The perimeter of triangle ABC is 42cm because AB + BC + AC = 6 + 8 + 28 = 42.
- The area of triangle ABC is 84 sq cm because the area of a triangle can be calculated using the formula: Area = (1/2) * base * height. In this case, the base is BC and the height is the radius of the inscribed circle. Therefore, the area is (1/2) * 14 * 4 = 84 sq cm.
Therefore, options A, B, C, and D are correct.
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Community Answer
A circle of radius 4cm is inscribed in ΔABC, which touches the side B...
Given, inradius of ΔABC = 4cm,
s − b = 6 and s − c = 8
(i) r = (s − a) tan A/2 = (s − b) tan B/2
= (s − c) tan C/2
⇒ tan B/2 = 2/3
and tan C/2 = 1/2
Since, 
all are less than 1
all are less than 1Hence, the triangle is necessarily acute angled triangle.
(ii) Solved in option (i).
⇒ 2s = 3a
⇒ 2s = 42
∴ Perimeter = 42
(iv) Δ = Area of ΔABC = r × s = 84 sq cm
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A circle of radius 4cm is inscribed in ΔABC, which touches the side BC at D, if BD = 6cm and DC = 8cm, then which of the following options are correct ?a)The triangle is necessarily acute angled triangleb)tan A/2 = 4/7c)Perimeter of the ΔABC is 42cmd)Area of ΔABC is 84 sq cmCorrect answer is option 'A,B,C,D'. Can you explain this answer?
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A circle of radius 4cm is inscribed in ΔABC, which touches the side BC at D, if BD = 6cm and DC = 8cm, then which of the following options are correct ?a)The triangle is necessarily acute angled triangleb)tan A/2 = 4/7c)Perimeter of the ΔABC is 42cmd)Area of ΔABC is 84 sq cmCorrect answer is option 'A,B,C,D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A circle of radius 4cm is inscribed in ΔABC, which touches the side BC at D, if BD = 6cm and DC = 8cm, then which of the following options are correct ?a)The triangle is necessarily acute angled triangleb)tan A/2 = 4/7c)Perimeter of the ΔABC is 42cmd)Area of ΔABC is 84 sq cmCorrect answer is option 'A,B,C,D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A circle of radius 4cm is inscribed in ΔABC, which touches the side BC at D, if BD = 6cm and DC = 8cm, then which of the following options are correct ?a)The triangle is necessarily acute angled triangleb)tan A/2 = 4/7c)Perimeter of the ΔABC is 42cmd)Area of ΔABC is 84 sq cmCorrect answer is option 'A,B,C,D'. Can you explain this answer?.
A circle of radius 4cm is inscribed in ΔABC, which touches the side BC at D, if BD = 6cm and DC = 8cm, then which of the following options are correct ?a)The triangle is necessarily acute angled triangleb)tan A/2 = 4/7c)Perimeter of the ΔABC is 42cmd)Area of ΔABC is 84 sq cmCorrect answer is option 'A,B,C,D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A circle of radius 4cm is inscribed in ΔABC, which touches the side BC at D, if BD = 6cm and DC = 8cm, then which of the following options are correct ?a)The triangle is necessarily acute angled triangleb)tan A/2 = 4/7c)Perimeter of the ΔABC is 42cmd)Area of ΔABC is 84 sq cmCorrect answer is option 'A,B,C,D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A circle of radius 4cm is inscribed in ΔABC, which touches the side BC at D, if BD = 6cm and DC = 8cm, then which of the following options are correct ?a)The triangle is necessarily acute angled triangleb)tan A/2 = 4/7c)Perimeter of the ΔABC is 42cmd)Area of ΔABC is 84 sq cmCorrect answer is option 'A,B,C,D'. Can you explain this answer?.
Solutions for A circle of radius 4cm is inscribed in ΔABC, which touches the side BC at D, if BD = 6cm and DC = 8cm, then which of the following options are correct ?a)The triangle is necessarily acute angled triangleb)tan A/2 = 4/7c)Perimeter of the ΔABC is 42cmd)Area of ΔABC is 84 sq cmCorrect answer is option 'A,B,C,D'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
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